Tugas Anreal (teorema 3.3.4 - 3.4.11)

Dian Novitasari (107017000745) Nur Malitasari (107017000734) Zulkahfi (107017000468) TEOREMA 3.3.4

{ } { } Jika barisan x n n =1 konvergen ke l . .maka setiap barisan bagian dari x n n =1 juga ∝

∝

konvergen ke l . . Ilustrasi : ∝

 2n   4 3 8    1, , , ... 1. χ =  n + 1 n =1 =  3 2 5  konvergen ke 2 ∝

 2n  2 6 4     ,1, , ... n + 2   n = 1 K= = 3 5 3  K merupakan barisan bagian dari χ, maka K juga konvergen ke 2 ∝

 n2  8   1 − ,−2,3, ,...  2   n − 6  n =1  5 5  2. χ =  = konvergen ke 1 ∝  n   1 2 3    − ,− ,− ,−4...  K =  n − 5  n =1 =  4 3 2

K merupakan barisan bagian dari χ, maka K juga konvergen ke 1 ∝

 n  1 2 3 4    , , , ,... 2  3. χ = 1 + n  n =1 =  2 5 10 17  konvergen ke 0 ∝

 1  1 1 1 1   2   , , , ... n + 1   n = 1 K= =  2 5 10 17  K merupakan barisan bagian dari χ, maka K juga konvergen ke 0 ∝

 n2  3 16  1 ,−4,− ,−, ,...   2  3 − n  n =1  2 2 13  4. χ =  = konvergen ke -1 ∝

 n2   1 4 9 16  , ,− ,− ,...  2  7 − n  n =1  K=  = 6 3 2 9  K merupakan barisan bagian dari χ, maka K juga konvergen ke -1

∝

 1  1 1 1 1     , , , ,... n + 3   n = 1 5. χ = = 4 5 6 7  konvergen ke 0 ∝

 n  1 1 1 1     , , , ... K =  n − 5  n =1 =  2 5 10 17  K merupakan barisan bagian dari χ, maka K juga konvergen ke 0

TEOREMA 3.4.4 Jika barisan bilangan real 010009000003460700000000be0400000000be04000026060f007209574d464301000000000 0010011a50000000001000000500900000000000050090000010000006c0000000000000000 00000033000000120000000000000000000000c1040000ba01000020454d460000010050090 0000c00000001000000000000000000000000000000560500000003000040010000b3000000 00000000000000000000000000e2040038bb0200460000002c00000020000000454d462b014 001001c000000100000000210c0db0100000060000000600000004600000010030000040300 00454d462b224004000c000000000000001e4009000c00000000000000244001000c0000000 00000003040020010000000040000000000803f214007000c00000000000000084000055c02 0000500200000210c0db01000000000000000000000000000000000000000100000089504e4 70d0a1a0a0000000d494844520000003400000013080200000050b3b9320000000173524742 00aece1ce9000000097048597300000ec400000ec401952b0e1b000001d649444154484bcd55 bb8d83401085ebc1044e882d5f031460e9b26be2dc004eae844bec0640d46141014e1d801c3b7 10045e0dd99fdc272ccfaac93574262613e6fe7cddb09fbbe0f5e75bdbd2a308e4b836bf290af5d d5f9e3edaa1d38e78dbfef6f1e8c56b6ea8cd964356e1e5d83206d9962621e17fec90d3541a0b0f d151a0b0470649c3a4bcb7674f2b6ccc4570242a0b5bb5d9e42c762b932e3ac960b153d4bd6fe 2900dc20a87f14e1c10f99c611eed69f71018d181d3fda3e39c17b185d930d40262da82ee74392 6034a1fc060d63ff7792c2ed9ed01d2a36aab5bd1e0241021051ed4e09477c389e3ba662b6e9fb fdfc89a3380d2eb707d43e5546d091bb2a283742c5cc324e8723286060a2d5ea00e14bd3b83bf c01991e482be763b4baaad8065e340dba83d4f3b6119f04fa2677f61cb6dd4ff02dda8eed72cae0 a0f49c1c26285e6328c1842a2e5695a458b5cc4c6d8a5b5e518e7ba648bcf187ad60f4c18ce504 dfdc4bebdd71cfadbfb436179b3d0f2395da9ca19adb7710b01a49921ae3329fb124326e5c6ba4 2b4a9f8d61b33d1cb3150d4c4b3571353ea3fe76e550ad22828ba961fd55227ca94b3e2a65460 bacd37246be53e068aa6719c54191d4e96ad32dedea40453147c81e6203fcbfd91de653257dd3 647cb90000000049454e44ae4260820000000840010824000000180000000210c0db0100000 0030000000000000000000000000000001b4000004000000034000000010000000200000000 0000bf000000bf000050420000984103000000000080b3000080b3ffff4f42000080b3000080b 3ffff97412100000008000000620000000c00000001000000150000000c00000004000000150 000000c0000000400000051000000dc04000000000000000000003300000012000000000000 00000000000000000000000000340000001300000050000000b000000000010000dc0300000

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7200ba729b009bbaea00ffeaba00d49b9b00eaffff000072ba0072007200eaba7200720000009b 72ba00eaffd4007272ba00eaeaba00eaba9b00727200009b007200babad400baeaea0072009b00 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010101010101010101010101010101010101010b21110102000801 0101010101010b2111010200080101010101010101010102000000160101010102000304010 10101010101010102080102081f040500060701010102080102081f04010101010101010101 010d1e07010101010206070101010101010101010101050a01090a010101010206070101050 a01090a01010d0e00000000171101010d1e070101010b0c0801010101010101010101011f200 e0a0b0f040101010d0e08011f200e0a0b0f0401010101010101010101010d1e070101010d0e08 01010217110105000304010d0e0a090003040101010b0c08010d0e0a0900030401010101010 101010101010d1e070101010b0c08010101131a1b14160d1c01010101010101010101010d15 110101010101010101010d0e0000000017110b0c1d1e0701010113141601010101131417110 101010101010101010101010109060701010101010101010101010101010101010101090607 0101010b0c160101010101090607010101010101010101010101010d1511010101010101010 101010101010101010101010101010101091711010101010101050016010101010101010101 01010101010b0c160101010101010101010101010101010101010101010101010b0c0801011 31819141602080101010101010101010101010d151101010101010101010101010101010101 0101010101010101011314160101010b0c17110102171101010101010101010101090607010 101010101010101010101010101010101010101010101010b0c080101010101010101010101 010101010101010101010d15110101050000160900000a01010101010101010101010101010 10d0e080101010101010101010101010101010101010101010b0c08010b120113140a010110 1101010101010101010101010101010b0c08010101010101010101010101010101010101010 1010d0e08010b0f040102060701101101010101010101010101010101010102060701010101 010101010101010101010101010101010206070101050000080900000a01010101010101010 101010101010101010200030401010101010101010101010101010101050006070101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010101010101010101010101010101010101010101010101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010c00000040092900aa0000000000000013003400000000000400 konvergen, maka 00002701ffff030000000000 010009000003460700000000be0400000000be04000026060f007209574d464301000000000 0010011a50000000001000000500900000000000050090000010000006c0000000000000000 00000033000000120000000000000000000000c1040000ba01000020454d460000010050090 0000c00000001000000000000000000000000000000560500000003000040010000b3000000 00000000000000000000000000e2040038bb0200460000002c00000020000000454d462b014 001001c000000100000000210c0db0100000060000000600000004600000010030000040300 00454d462b224004000c000000000000001e4009000c00000000000000244001000c0000000 00000003040020010000000040000000000803f214007000c00000000000000084000055c02 0000500200000210c0db01000000000000000000000000000000000000000100000089504e4 70d0a1a0a0000000d494844520000003400000013080200000050b3b9320000000173524742

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0101010101010101010101010101010101010101010b0c08010101010101010101010101010 1010101010101010d15110101050000160900000a0101010101010101010101010101010d0e 080101010101010101010101010101010101010101010b0c08010b120113140a01011011010 10101010101010101010101010b0c080101010101010101010101010101010101010101010d 0e08010b0f04010206070110110101010101010101010101010101010206070101010101010 1010101010101010101010101010206070101050000080900000a0101010101010101010101 010101010101020003040101010101010101010101010101010105000607010101010101010 101010101010101010101010101010101010101010101010101010101010101010101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010101010101010101010101010101010101010101010101010101 0101010101010101014c0000006400000000000000000000003300000012000000000000000 000000034000000130000002900aa0000000000000000000000803f00000000000000000000 803f00000000000000000000000000000000000000000000000000000000000000002200000 00c000000ffffffff460000001c00000010000000454d462b024000000c000000000000000e000 000140000000000000010000000140000000400000003010800050000000b02000000000500 00000c0213003400030000001e000400000007010400040000000701040053020000410b200 0cc001300340000000000130034000000000028000000340000001300000001000800000000 00000000000000000000000000220000000000000000000000ffffff00009bd400ba720000ffff ea0072baea009b000000ffffd400ffd49b009bd4ff00d49b0000d4ffff0000009b00baeaff000000 7200ba729b009bbaea00ffeaba00d49b9b00eaffff000072ba0072007200eaba7200720000009b 72ba00eaffd4007272ba00eaeaba00eaba9b00727200009b007200babad400baeaea0072009b00 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010101010101010101010101010101010101010b21110102000801 0101010101010b2111010200080101010101010101010102000000160101010102000304010 10101010101010102080102081f040500060701010102080102081f04010101010101010101 010d1e07010101010206070101010101010101010101050a01090a010101010206070101050 a01090a01010d0e00000000171101010d1e070101010b0c0801010101010101010101011f200 e0a0b0f040101010d0e08011f200e0a0b0f0401010101010101010101010d1e070101010d0e08 01010217110105000304010d0e0a090003040101010b0c08010d0e0a0900030401010101010 101010101010d1e070101010b0c08010101131a1b14160d1c01010101010101010101010d15 110101010101010101010d0e0000000017110b0c1d1e0701010113141601010101131417110 101010101010101010101010109060701010101010101010101010101010101010101090607 0101010b0c160101010101090607010101010101010101010101010d1511010101010101010 101010101010101010101010101010101091711010101010101050016010101010101010101 01010101010b0c160101010101010101010101010101010101010101010101010b0c0801011 31819141602080101010101010101010101010d151101010101010101010101010101010101 0101010101010101011314160101010b0c17110102171101010101010101010101090607010 101010101010101010101010101010101010101010101010b0c080101010101010101010101 010101010101010101010d15110101050000160900000a01010101010101010101010101010 10d0e080101010101010101010101010101010101010101010b0c08010b120113140a010110

1101010101010101010101010101010b0c08010101010101010101010101010101010101010 1010d0e08010b0f040102060701101101010101010101010101010101010102060701010101 010101010101010101010101010101010206070101050000080900000a01010101010101010 101010101010101010200030401010101010101010101010101010101050006070101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010101010101010101010101010101010101010101010101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101010c00000040092900aa0000000000000013003400000000000400 terbatas. 00002701ffff030000000000 Contoh : ∝

∝

1 1     Jika  3n  n =1 konvergen ke 0, maka  3n  n =1 terbatas.

Bukti ; ∝

1 1 1 1 1  1    , , , ...  3n  n =1 =  3 6 9 12  terbatas di atas oleh 0 dan terbatas di bawah oleh 3 ∝

∝

1  1   2 + 1  2 + 1  n =1 konvergen ke 0, maka  n  n =1 terbatas. 1. Jika  n

Bukti ; ∝

1   5 10 26   2 + 1 2, , , ... n  n =1 =  4 9 25  terbatas di atas oleh 1 dan terbatas di bawah oleh 2 ∝

∝

1 1   1 −  1 −  2. Jika  n  n =1 konvergen ke 0, maka  n  n =1 terbatas.

Bukti ; ∝

1   1 2 3  1 −  0, , , ...  n  n =1 =  2 3 4  terbatas di atas oleh 1 dan terbatas di bawah oleh 0 ∝

∝

 2n   2n      3. Jika  n + 1 n =1 konvergen ke 0, maka  n + 1 n =1 terbatas.

Bukti ; ∝

 2n   4 6 8    1, , , ...  n + 1 n =1 =  3 4 5  terbatas di atas oleh 2 dan terbatas di bawah oleh 1

∝

∝

 1   1      4. Jika  n  n =1 konvergen ke 0, maka  n  n =1 terbatas.

Bukti ; ∝

 1     n  n =1 =

 1 1 1  , , ... 1,  2 3 2  terbatas di atas oleh 0 dan terbatas di bawah oleh 1

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∝

 n    Karena  2n + 1 n =1 barisan tak turun dan terbatas di atas, maka konvergen.

∝

 n     2n + 1 n =1

∝

 3n   6 9 12    1, , , ... 2. Misalkan  2n + 1 n =1 =  5 7 9  terbatas atas oleh 2. ∝

∝

 3n   3n      Karena  2n + 1 n =1 barisan tak turun dan terbatas di atas, maka  2n + 1 n =1 konvergen.

∝

 4n    n + 5   n =1 = 3. Misalkan

 1 8 3 16   , , , ...  6 7 2 9  terbatas atas oleh 1.

∝

∝

 4n   4n      n + 5 n + 5     n =1 n = 1 Karena barisan tak turun dan terbatas di atas, maka

konvergen.

∝

 n2  1 2 9 8   2   , , , ... n + 2   n = 1 4. Misalkan =  3 3 11 9  terbatas atas oleh 1. ∝

∝

 n2   n2   2   2  n + 2  n =1 n + 2  n =1 Karena  barisan tak turun dan terbatas di atas, maka  konvergen.

∝

 n2 + 3  3 7 6 14  , , , ...  2  n + 5  n =1  5. Misalkan  =  5 9 7 15  terbatas atas oleh 1. ∝

 n2 + 3  2  n + 5  n =1  Karena barisan tak turun dan terbatas di atas, maka konvergen.

∝

 n2 + 3  2   n + 5  n =1

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00000000000000000000000000e2040038bb0200460000002c00000020000000454d462b014 001001c000000100000000210c0db01000000600000006000000046000000d0010000c40100 00454d462b224004000c000000000000001e4009000c00000000000000244001000c0000000 00000003040020010000000040000000000803f214007000c00000000000000084000051c01 0000100100000210c0db01000000000000000000000000000000000000000100000089504e4 70d0a1a0a0000000d494844520000000e0000001308020000002f62abc300000001735247420 0aece1ce9000000097048597300000ec400000ec401952b0e1b0000009949444154384f63fcfff f3f0371808938652055434fe9ebddc58c1030e72ac49f98220c0cc0c0ba329ba168d72b2003cc0 6325fed2a02aa9e7d05c80731a19240a54095605128007261eac02240c510b54c0caf9f5d2f921 74304af983cc81c7f6d9888a89466dfc357e070853321725737baf53140e4a0ae861b85e256b02 3218e85b9096eff7f90b7208e870024ff4145e0fe601cf1c905004d0898fed59c2847000000004 9454e44ae4260820840010824000000180000000210c0db0100000003000000000000000000 0000000000001b40000040000000340000000100000002000000000000bf000000bf0000604 10000984103000000000000b3000000b3ffff5f41000000b3000000b3ffff97412100000008000 000620000000c00000001000000150000000c00000004000000150000000c00000004000000 510000000802000000000000000000000d00000012000000000000000000000000000000000 000000e000000130000005000000088000000d800000030010000000000002000cc000e0000 0013000000280000000e0000001300000001000800000000000000000000000000000000001 80000000000000000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400e aeaba009b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00b a727200ffffea009bd4ff00d49b0000000072007272ba00010101010101010101010101010100 000101010101010101010101010101000001010101010101010101010101010000010101010 101010101010101010100000101010101010101010101010101064401010101010101010101 010101010b4401020300000601010500000401010000010506010b160611040102170d01000 00e0f0a010114001501010111060100000b1213010102030601010114150100000e0f0a01010 e10090a010111060100000105060101070803090a0b0c0d0100000102030000040102030000 0401010000010101010101010101010101010100000101010101010101010101010101d8410 1010101010101010101010101010f0501010101010101010101010101010000010101010101 01010101010101010000010101010101010101010101010100004c000000640000000000000 0000000000d0000001200000000000000000000000e000000130000002900aa000000000000 0000000000803f00000000000000000000803f0000000000000000000000000000000000000 000000000000000000000000000220000000c000000ffffffff460000001c00000010000000454 d462b024000000c000000000000000e00000014000000000000001000000014000000040000 0003010800050000000b0200000000050000000c0213000e00030000001e000400000007010 4000400000007010400e9000000410b2000cc0013000e000000000013000e00000000002800 00000e000000130000000100080000000000000000000000000000000000180000000000000 000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400eaeaba009b00000 0ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00ba727200ffffea00 9bd4ff00d49b0000000072007272ba0001010101010101010101010101010000010101010101 010101010101010100000101010101010101010101010101000001010101010101010101010

1010100000101010101010101010101010101064401010101010101010101010101010b4401 020300000601010500000401010000010506010b160611040102170d0100000e0f0a0101140 01501010111060100000b1213010102030601010114150100000e0f0a01010e10090a0101110 60100000105060101070803090a0b0c0d010000010203000004010203000004010100000101 01010101010101010101010100000101010101010101010101010101d841010101010101010 10101010101010f050101010101010101010101010101000001010101010101010101010101 010000010101010101010101010101010100000c00000040092900aa0000000000000013000 e0000000000040000002701ffff030000000000. Contoh : 1.

{n }

2 ∝ n =1

= {1, 4, 9, 16,..}

{ }

2 Merupakan barisan tak terbatas diatas karena n

{ }

2 terbatas diatas, maka n

∝ n =1

barisan tak turun dan tak

∝ n =1

divergen ke +

010009000003d20300000000b40200000000b402000026060f005e05574d464301000 00000000100b87300000000010000003c050000000000003c050000010000006c0000 0000000000000000000d00000012000000000000000000000047010000ba010000204 54d46000001003c0500000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000d0010000c4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000051c010000100100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000000e0000001308020000002f62abc3000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b0000009949444154384f63fcffff3f 0371808938652055434fe9ebddc58c1030e72ac49f98220c0cc0c0ba329ba168d72b200 3cc06325fed2a02aa9e7d05c80731a19240a54095605128007261eac02240c510b54c0c af9f5d2f92174304af983cc81c7f6d9888a89466dfc357e070853321725737baf53140e4 a0ae861b85e256b023218e85b9096eff7f90b7208e870024ff4145e0fe601cf1c905004d 0898fed59c28470000000049454e44ae4260820840010824000000180000000210c0db 01000000030000000000000000000000000000001b400000400000003400000001000 00002000000000000bf000000bf000060410000984103000000000000b3000000b3ffff 5f41000000b3000000b3ffff97412100000008000000620000000c00000001000000150 000000c00000004000000150000000c00000004000000510000000802000000000000 000000000d00000012000000000000000000000000000000000000000e00000013000 0005000000088000000d800000030010000000000002000cc000e0000001300000028

0000000e0000001300000001000800000000000000000000000000000000001800000 00000000000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400e aeaba009b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072b aea00ba727200ffffea009bd4ff00d49b0000000072007272ba000101010101010101010 101010101000001010101010101010101010101010000010101010101010101010101 010100000101010101010101010101010101000001010101010101010101010101010 64401010101010101010101010101010b440102030000060101050000040101000001 0506010b160611040102170d0100000e0f0a010114001501010111060100000b12130 10102030601010114150100000e0f0a01010e10090a010111060100000105060101070 803090a0b0c0d01000001020300000401020300000401010000010101010101010101 010101010100000101010101010101010101010101d84101010101010101010101010 101010f05010101010101010101010101010100000101010101010101010101010101 0000010101010101010101010101010100004c0000006400000000000000000000000 d0000001200000000000000000000000e000000130000002900aa0000000000000000 000000803f00000000000000000000803f00000000000000000000000000000000000 00000000000000000000000000000220000000c000000ffffffff460000001c000000100 00000454d462b024000000c000000000000000e000000140000000000000010000000 140000000400000003010800050000000b0200000000050000000c0213000e0003000 0001e0004000000070104000400000007010400e9000000410b2000cc0013000e0000 00000013000e0000000000280000000e0000001300000001000800000000000000000 00000000000000000180000000000000000000000ffffff00eaffff000072ba00ffd49b00 009bd400eaba7200729bd400eaeaba009b000000ffffd400baeaff0072007200ffeaba00d 4ffff009b009b0000009b0072baea00ba727200ffffea009bd4ff00d49b00000000720072 72ba00010101010101010101010101010100000101010101010101010101010101000 001010101010101010101010101010000010101010101010101010101010100000101 010101010101010101010101064401010101010101010101010101010b44010203000 00601010500000401010000010506010b160611040102170d0100000e0f0a01011400 1501010111060100000b1213010102030601010114150100000e0f0a01010e10090a01 0111060100000105060101070803090a0b0c0d0100000102030000040102030000040 1010000010101010101010101010101010100000101010101010101010101010101d8 4101010101010101010101010101010f0501010101010101010101010101010000010 10101010101010101010101010000010101010101010101010101010100000c000000 40092900aa0000000000000013000e0000000000040000002701ffff030000000000

2.

{ n + 5} ∝n =1

= {6, 7, 8, 9,..}

{ } Merupakan barisan tak terbatas diatas karena n + 5 n =1 barisan tak turun dan tak ∝

{ } terbatas diatas, maka n + 5 n =1 divergen ke + ∝

010009000003d20300000000b40200000000b402000026060f005e05574d464301000 00000000100b87300000000010000003c050000000000003c050000010000006c0000 0000000000000000000d00000012000000000000000000000047010000ba010000204 54d46000001003c0500000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000d0010000c4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000051c010000100100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000000e0000001308020000002f62abc3000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b0000009949444154384f63fcffff3f 0371808938652055434fe9ebddc58c1030e72ac49f98220c0cc0c0ba329ba168d72b200 3cc06325fed2a02aa9e7d05c80731a19240a54095605128007261eac02240c510b54c0c af9f5d2f92174304af983cc81c7f6d9888a89466dfc357e070853321725737baf53140e4 a0ae861b85e256b023218e85b9096eff7f90b7208e870024ff4145e0fe601cf1c905004d 0898fed59c28470000000049454e44ae4260820840010824000000180000000210c0db 01000000030000000000000000000000000000001b400000400000003400000001000 00002000000000000bf000000bf000060410000984103000000000000b3000000b3ffff 5f41000000b3000000b3ffff97412100000008000000620000000c00000001000000150 000000c00000004000000150000000c00000004000000510000000802000000000000 000000000d00000012000000000000000000000000000000000000000e00000013000 0005000000088000000d800000030010000000000002000cc000e0000001300000028 0000000e0000001300000001000800000000000000000000000000000000001800000 00000000000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400e aeaba009b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072b aea00ba727200ffffea009bd4ff00d49b0000000072007272ba000101010101010101010 101010101000001010101010101010101010101010000010101010101010101010101 010100000101010101010101010101010101000001010101010101010101010101010 64401010101010101010101010101010b440102030000060101050000040101000001 0506010b160611040102170d0100000e0f0a010114001501010111060100000b12130

10102030601010114150100000e0f0a01010e10090a010111060100000105060101070 803090a0b0c0d01000001020300000401020300000401010000010101010101010101 010101010100000101010101010101010101010101d84101010101010101010101010 101010f05010101010101010101010101010100000101010101010101010101010101 0000010101010101010101010101010100004c0000006400000000000000000000000 d0000001200000000000000000000000e000000130000002900aa0000000000000000 000000803f00000000000000000000803f00000000000000000000000000000000000 00000000000000000000000000000220000000c000000ffffffff460000001c000000100 00000454d462b024000000c000000000000000e000000140000000000000010000000 140000000400000003010800050000000b0200000000050000000c0213000e0003000 0001e0004000000070104000400000007010400e9000000410b2000cc0013000e0000 00000013000e0000000000280000000e0000001300000001000800000000000000000 00000000000000000180000000000000000000000ffffff00eaffff000072ba00ffd49b00 009bd400eaba7200729bd400eaeaba009b000000ffffd400baeaff0072007200ffeaba00d 4ffff009b009b0000009b0072baea00ba727200ffffea009bd4ff00d49b00000000720072 72ba00010101010101010101010101010100000101010101010101010101010101000 001010101010101010101010101010000010101010101010101010101010100000101 010101010101010101010101064401010101010101010101010101010b44010203000 00601010500000401010000010506010b160611040102170d0100000e0f0a01011400 1501010111060100000b1213010102030601010114150100000e0f0a01010e10090a01 0111060100000105060101070803090a0b0c0d0100000102030000040102030000040 1010000010101010101010101010101010100000101010101010101010101010101d8 4101010101010101010101010101010f0501010101010101010101010101010000010 10101010101010101010101010000010101010101010101010101010100000c000000 40092900aa0000000000000013000e0000000000040000002701ffff030000000000

3.

{ 2n + 1} ∝n=1

= {3, 5, 7, 9,..}

{ } Merupakan barisan tak terbatas diatas karena 2n + 1 n =1 barisan tak turun dan tak ∝

{ } terbatas diatas, maka 2n + 1 n =1 divergen ke + ∝

010009000003d20300000000b40200000000b402000026060f005e05574d464301000 00000000100b87300000000010000003c050000000000003c050000010000006c0000 0000000000000000000d00000012000000000000000000000047010000ba010000204 54d46000001003c0500000c0000000100000000000000000000000000000056050000

4.

{ 3n} ∝n=1

= {3, 6, 9, 12,..}

{ } Merupakan barisan tak terbatas diatas karena 3n n =1 barisan tak turun dan tak ∝

{ } terbatas diatas, maka 3n n =1 divergen ke + ∝

010009000003d20300000000b40200000000b402000026060f005e05574d464301000 00000000100b87300000000010000003c050000000000003c050000010000006c0000 0000000000000000000d00000012000000000000000000000047010000ba010000204 54d46000001003c0500000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000d0010000c4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000051c010000100100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000000e0000001308020000002f62abc3000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b0000009949444154384f63fcffff3f

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001010101010101010101010101010000010101010101010101010101010100000101 010101010101010101010101064401010101010101010101010101010b44010203000 00601010500000401010000010506010b160611040102170d0100000e0f0a01011400 1501010111060100000b1213010102030601010114150100000e0f0a01010e10090a01 0111060100000105060101070803090a0b0c0d0100000102030000040102030000040 1010000010101010101010101010101010100000101010101010101010101010101d8 4101010101010101010101010101010f0501010101010101010101010101010000010 10101010101010101010101010000010101010101010101010101010100000c000000 40092900aa0000000000000013000e0000000000040000002701ffff030000000000

5.

{n

2

}

+2

∝ n =1

= {3, 6, 11, 18,..}

Merupakan barisan tak terbatas diatas karena {n terbatas diatas, maka {n + 2} divergen ke + 2

2

}

+2

∝ n =1

barisan tak turun dan tak

∝

n =1

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∝

1   − 1 karena  n  n =1 barisan tak naik dan terbatas di bawah maka ∝

 1     n + 2  n =1 =

∝

1   − 1  n  n =1 konvergen

1 1 1 1   , , , ...  3 4 5 6  terbatas di bawah oleh 0

2. ∝

∝

 1   1      karena  n + 2  n =1 barisan tak naik dan terbatas di bawah maka  n + 2  n =1 konvergen ∝

1 1 1 1 1     , , , ...  2n  n =1 =  2 4 6 8  terbatas di bawah oleh 0

3. ∝

∝

1 1     karena  2n  n =1 barisan tak naik dan terbatas di bawah maka  2n  n =1 konvergen ∝

 1   1 1 1    − ,− ,− ,−1...  n − 5  n =1 =  4 3 2  terbatas di bawah oleh 0

4. ∝

∝

 1   1      karena  n − 5  n =1 barisan tak naik dan terbatas di bawah maka  n − 5  n =1 konvergen ∝

1   5 11 17   − 3 − ,− ,− ... 6  terbatas di bawah oleh -3  2n  n =1 =  3 4

5. ∝

1   − 3 karena  2n  n =1 barisan tak naik dan terbatas di bawah maka

∝

1   − 3  2n  n =1 konvergen

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1040004000000070104005d010000410b2000cc0013001a000000000013001a000000000028 0000001a0000001300000001000800000000000000000000000000000000001900000000000 00000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400eaeaba009b000 000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00ba720000ffffea 00ba7272009bd4ff00d49b0000000072007272ba000101010101010101010101010101010101 010101010101010101000001010101010101010101010101010101010101010101010101010 000010101010101010101010101010101010101010101010101010100000101010101010101 010101010101010101010101010101010101000001010101010101010101010101010101010 101010101010101010000010101010101010101010101010101010101010101010101010100 000101010101010101010101010102030000060101050000040101000001010101010101010 1010101010506010b170611040102180d0100000101010101010101010101010e0f0a010115 001601010111060100000203000000000000000012130b14130101020306010101151601000 00101010101010101010101010e0f0a01010e10090a010111060100000101010101010101010 101010105060101070803090a0b0c0d01000001010101010101010101010101020300000401 020300000401010000010101010101010101010101010101010101010101010101010100000 101010101010101010101010101010101010101010101010101000001010101010101010101 010101010101010101010101010101010000010101010101010101010101010101010101010 101010101010100000101010101010101010101010101010101010101010101010101000001 0101010101010101010101010101010101010101010101010100000c00000040092900aa000 0000000000013001a0000000000040000002701ffff030000000000. Contoh : 1.

{4 − n }

2 ∝ n =1

= {3, 0, -5,-12, …..}

Merupakan barisan tak terbatas dibawah karena { 4 − n } terbatas dibawah, maka { 4 − n } divergen ke

2 ∝ n =1

barisan tak naik dan tak

2 ∝ n =1

010009000003ca04000000003803000000003803000026060f006606574d464301000 00000000100a0910000000001000000440600000000000044060000010000006c0000 0000000000000000001900000012000000000000000000000060020000ba010000204 54d4600000100440600000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000f0010000e4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000053c010000300100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000001a00000013080200000001278a51000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b000000b949444154384f63fcffff3f

2.

{ − 3n} ∝n =1

= {-3, -6, -9,-12, …..}

Merupakan barisan tak terbatas dibawah karena { − 3n terbatas dibawah, maka { − 3n

}

}

∝ n =1

barisan tak naik dan tak

∝ n =1

divergen ke

010009000003ca04000000003803000000003803000026060f006606574d464301000 00000000100a0910000000001000000440600000000000044060000010000006c0000 0000000000000000001900000012000000000000000000000060020000ba010000204 54d4600000100440600000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000

0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000f0010000e4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000053c010000300100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000001a00000013080200000001278a51000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b000000b949444154384f63fcffff3f 03f50013f58c0299346a1cf9e1497cd8bdde5dcc080173ae42ecc314616000261462c09 5d90c45bb5e41545e990d64beda55043471f615201fc4844a12691cd034b04e2800726 1668145800642cc037a16e166a86790a9e2ddaf41de7a76bd485e0c11a462f220f7f86bc 34444a534fb1ebe02f188f129c87fc8ae43771cc275441a07320f1a3ce04083041eccff70 bffe27d63848804300529c4045e0e1ca385a04909dcb88cf6444594165e300ac1c66c13f 29e0f50000000049454e44ae4260820840010824000000180000000210c0db01000000 030000000000000000000000000000001b40000040000000340000000100000002000 000000000bf000000bf0000d0410000984103000000000080b3000080b3ffffcf410000 80b3000080b3ffff97412100000008000000620000000c00000001000000150000000c 00000004000000150000000c0000000400000051000000f0020000000000000000000 01900000012000000000000000000000000000000000000001a000000130000005000 00008c000000dc00000014020000000000002000cc001a00000013000000280000001 a00000013000000010008000000000000000000000000000000000019000000000000 0000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400eaeaba00 9b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00ba 720000ffffea00ba7272009bd4ff00d49b0000000072007272ba0001010101010101010 101010101010101010101010101010101010000010101010101010101010101010101 010101010101010101010100000101010101010101010101010101010101010101010 101010101000001010101010101010101010101010101010101010101010101010000 010101010101010101010101010101010101010101010101010100000101010101010 101010101010101010101010101010101010101000001010101010101010101010101 020300000601010500000401010000010101010101010101010101010506010b17061 1040102180d0100000101010101010101010101010e0f0a0101150016010101110601 00000203000000000000000012130b141301010203060101011516010000010101010 1010101010101010e0f0a01010e10090a0101110601000001010101010101010101010 10105060101070803090a0b0c0d010000010101010101010101010101010203000004 010203000004010100000101010101010101010101010101010101010101010101010 101000001010101010101010101010101010101010101010101010101010000010101

010101010101010101010101010101010101010101010100000101010101010101010 101010101010101010101010101010101000001010101010101010101010101010101 010101010101010101010000010101010101010101010101010101010101010101010 101010100004c00000064000000000000000000000019000000120000000000000000 0000001a000000130000002900aa0000000000000000000000803f000000000000000 00000803f000000000000000000000000000000000000000000000000000000000000 0000220000000c000000ffffffff460000001c00000010000000454d462b024000000c00 0000000000000e0000001400000000000000100000001400000004000000030108000 50000000b0200000000050000000c0213001a00030000001e00040000000701040004 000000070104005d010000410b2000cc0013001a000000000013001a0000000000280 000001a00000013000000010008000000000000000000000000000000000019000000 0000000000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400ea eaba009b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072ba ea00ba720000ffffea00ba7272009bd4ff00d49b0000000072007272ba00010101010101 010101010101010101010101010101010101010100000101010101010101010101010 101010101010101010101010101000001010101010101010101010101010101010101 010101010101010000010101010101010101010101010101010101010101010101010 100000101010101010101010101010101010101010101010101010101000001010101 010101010101010101010101010101010101010101010000010101010101010101010 10101020300000601010500000401010000010101010101010101010101010506010b 170611040102180d0100000101010101010101010101010e0f0a01011500160101011 1060100000203000000000000000012130b1413010102030601010115160100000101 010101010101010101010e0f0a01010e10090a01011106010000010101010101010101 0101010105060101070803090a0b0c0d0100000101010101010101010101010102030 000040102030000040101000001010101010101010101010101010101010101010101 010101010000010101010101010101010101010101010101010101010101010100000 101010101010101010101010101010101010101010101010101000001010101010101 010101010101010101010101010101010101010000010101010101010101010101010 101010101010101010101010100000101010101010101010101010101010101010101 01010101010100000c00000040092900aa0000000000000013001a000000000004000 0002701ffff030000000000

3.

{ 5 − 2n} ∝n =1

= {3, 1, -1,-3, …..}

Merupakan barisan tak terbatas dibawah karena { 5 − 2n} n =1 barisan tak naik dan tak ∝

terbatas dibawah, maka { 5 − 2n} n =1 divergen ke ∝

010009000003ca04000000003803000000003803000026060f006606574d464301000 00000000100a0910000000001000000440600000000000044060000010000006c0000 0000000000000000001900000012000000000000000000000060020000ba010000204 54d4600000100440600000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000f0010000e4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000053c010000300100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000001a00000013080200000001278a51000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b000000b949444154384f63fcffff3f 03f50013f58c0299346a1cf9e1497cd8bdde5dcc080173ae42ecc314616000261462c09 5d90c45bb5e41545e990d64beda55043471f615201fc4844a12691cd034b04e2800726 1668145800642cc037a16e166a86790a9e2ddaf41de7a76bd485e0c11a462f220f7f86bc 34444a534fb1ebe02f188f129c87fc8ae43771cc275441a07320f1a3ce04083041eccff70 bffe27d63848804300529c4045e0e1ca385a04909dcb88cf6444594165e300ac1c66c13f 29e0f50000000049454e44ae4260820840010824000000180000000210c0db01000000 030000000000000000000000000000001b40000040000000340000000100000002000 000000000bf000000bf0000d0410000984103000000000080b3000080b3ffffcf410000 80b3000080b3ffff97412100000008000000620000000c00000001000000150000000c 00000004000000150000000c0000000400000051000000f0020000000000000000000 01900000012000000000000000000000000000000000000001a000000130000005000 00008c000000dc00000014020000000000002000cc001a00000013000000280000001 a00000013000000010008000000000000000000000000000000000019000000000000 0000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400eaeaba00 9b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00ba 720000ffffea00ba7272009bd4ff00d49b0000000072007272ba0001010101010101010 101010101010101010101010101010101010000010101010101010101010101010101 010101010101010101010100000101010101010101010101010101010101010101010 101010101000001010101010101010101010101010101010101010101010101010000 010101010101010101010101010101010101010101010101010100000101010101010 101010101010101010101010101010101010101000001010101010101010101010101 020300000601010500000401010000010101010101010101010101010506010b17061

4.

{1 − n }

∝ n =1

= {0, -1, -2,-3, …..}

Merupakan barisan tak terbatas dibawah karena {1 − n } terbatas dibawah, maka {1 − n } divergen ke

∝ n =1

barisan tak naik dan tak

∝

n =1

010009000003ca04000000003803000000003803000026060f006606574d464301000 00000000100a0910000000001000000440600000000000044060000010000006c0000 0000000000000000001900000012000000000000000000000060020000ba010000204 54d4600000100440600000c0000000100000000000000000000000000000056050000 0003000040010000b300000000000000000000000000000000e2040038bb020046000 0002c00000020000000454d462b014001001c000000100000000210c0db0100000060 0000006000000046000000f0010000e4010000454d462b224004000c0000000000000 01e4009000c00000000000000244001000c0000000000000030400200100000000400 00000000803f214007000c00000000000000084000053c010000300100000210c0db0 1000000000000000000000000000000000000000100000089504e470d0a1a0a000000 0d494844520000001a00000013080200000001278a51000000017352474200aece1ce9 000000097048597300000ec400000ec401952b0e1b000000b949444154384f63fcffff3f 03f50013f58c0299346a1cf9e1497cd8bdde5dcc080173ae42ecc314616000261462c09 5d90c45bb5e41545e990d64beda55043471f615201fc4844a12691cd034b04e2800726 1668145800642cc037a16e166a86790a9e2ddaf41de7a76bd485e0c11a462f220f7f86bc 34444a534fb1ebe02f188f129c87fc8ae43771cc275441a07320f1a3ce04083041eccff70 bffe27d63848804300529c4045e0e1ca385a04909dcb88cf6444594165e300ac1c66c13f 29e0f50000000049454e44ae4260820840010824000000180000000210c0db01000000 030000000000000000000000000000001b40000040000000340000000100000002000 000000000bf000000bf0000d0410000984103000000000080b3000080b3ffffcf410000 80b3000080b3ffff97412100000008000000620000000c00000001000000150000000c 00000004000000150000000c0000000400000051000000f0020000000000000000000 01900000012000000000000000000000000000000000000001a000000130000005000 00008c000000dc00000014020000000000002000cc001a00000013000000280000001 a00000013000000010008000000000000000000000000000000000019000000000000 0000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400eaeaba00 9b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072baea00ba

720000ffffea00ba7272009bd4ff00d49b0000000072007272ba0001010101010101010 101010101010101010101010101010101010000010101010101010101010101010101 010101010101010101010100000101010101010101010101010101010101010101010 101010101000001010101010101010101010101010101010101010101010101010000 010101010101010101010101010101010101010101010101010100000101010101010 101010101010101010101010101010101010101000001010101010101010101010101 020300000601010500000401010000010101010101010101010101010506010b17061 1040102180d0100000101010101010101010101010e0f0a0101150016010101110601 00000203000000000000000012130b141301010203060101011516010000010101010 1010101010101010e0f0a01010e10090a0101110601000001010101010101010101010 10105060101070803090a0b0c0d010000010101010101010101010101010203000004 010203000004010100000101010101010101010101010101010101010101010101010 101000001010101010101010101010101010101010101010101010101010000010101 010101010101010101010101010101010101010101010100000101010101010101010 101010101010101010101010101010101000001010101010101010101010101010101 010101010101010101010000010101010101010101010101010101010101010101010 101010100004c00000064000000000000000000000019000000120000000000000000 0000001a000000130000002900aa0000000000000000000000803f000000000000000 00000803f000000000000000000000000000000000000000000000000000000000000 0000220000000c000000ffffffff460000001c00000010000000454d462b024000000c00 0000000000000e0000001400000000000000100000001400000004000000030108000 50000000b0200000000050000000c0213001a00030000001e00040000000701040004 000000070104005d010000410b2000cc0013001a000000000013001a0000000000280 000001a00000013000000010008000000000000000000000000000000000019000000 0000000000000000ffffff00eaffff000072ba00ffd49b00009bd400eaba7200729bd400ea eaba009b000000ffffd400baeaff0072007200ffeaba00d4ffff009b009b0000009b0072ba ea00ba720000ffffea00ba7272009bd4ff00d49b0000000072007272ba00010101010101 010101010101010101010101010101010101010100000101010101010101010101010 101010101010101010101010101000001010101010101010101010101010101010101 010101010101010000010101010101010101010101010101010101010101010101010 100000101010101010101010101010101010101010101010101010101000001010101 010101010101010101010101010101010101010101010000010101010101010101010 10101020300000601010500000401010000010101010101010101010101010506010b 170611040102180d0100000101010101010101010101010e0f0a01011500160101011 1060100000203000000000000000012130b1413010102030601010115160100000101

010101010101010101010e0f0a01010e10090a01011106010000010101010101010101 0101010105060101070803090a0b0c0d0100000101010101010101010101010102030 000040102030000040101000001010101010101010101010101010101010101010101 010101010000010101010101010101010101010101010101010101010101010100000 101010101010101010101010101010101010101010101010101000001010101010101 010101010101010101010101010101010101010000010101010101010101010101010 101010101010101010101010100000101010101010101010101010101010101010101 01010101010100000c00000040092900aa0000000000000013001a000000000004000 0002701ffff030000000000

5.

{2 − n }

2 ∝ n =1

= {1, -2, -7,-14, …..}

2 Merupakan barisan tak terbatas dibawah karena { 2 − n } n =1 barisan tak naik dan tak ∝

2 terbatas dibawah, maka { 2 − n } n =1 divergen ke ∝

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0000000000000000220000000c000000ffffffff460000001c00000010000000454d462b024000 000c000000000000000e0000001400000000000000100000001400000004000000030108000 50000000b0200000000050000000c0213003500030000001e00040000000701040004000000 070104007d020000410b2000cc0013003500000000001300350000000000280000003500000 0130000000100080000000000000000000000000000000000240000000000000000000000fff fff00009bd400ba720000ffffea0072baea009b000000ffffd400ffd49b009bd4ff00d49b0000d4ff ff00ba729b009bbaea00ffeaba0000009b00baeaff0000007200d49b9b00eaffff000072ba00eaba 72007200000072007200d4d49b00baba9b00eaeaba00ba9bba007272ba009b00720072baba00 729bd40072720000babad400baeaea0072009b00010101010101010101010101010101010101 0101010101010101010101010101010101010101010101010101010101010101010101ffffff0 10101020003040101010101010101010b230e01020008010500060701010b230e0102000801 010101010101010101020000001501ffffff01010206070101010101010101010101010208010 2082104010102060701010208010208210401010101010101010101101d070101ffffff010b0f 0801010101010101010101010101050a01090a010101011011080101050a01090a010110110 0000000160e0101101d070101ffffff011011080101010101010101010101012122110a0b0c04 0101010b0f08012122110a0b0c040101010101010101010101101d070101ffffff010b0f080113 14160e0110110008010110110a0900030401010110170e0110110a090003040101010101010 101010101101d070101ffffff011314150101090006070503041f0e0101010101010101010101 090607010101010101010101101100000000160e0b0f201d070101ffffff010b0f150101010110 1d1e1501131b040101010101010101010110170e01010101010101010101010101010101010 1010906070101ffffff09160e0101010101010b0f150101010101010101010101010101010b0f1 50101010101010101010101010101010101010101010101ffffff010b0f080101131c0e131406 070101010101010101010101010110170e01010101010101010101010101010101010101010 1010101ffffff011314150101010b0f180f19171a1b04010101010101010101010906070101010 10101010101010101010101010101010101010101ffffff010b0f08010101010102160e131416 0e0101010101010101010110170e01010101010101010101010101010101010101010101010 1ffffff01101108010101010101010101010101010101010101010101010b0f08010105000015

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mempunyai barisan bagian yang monoton.

Contoh : 010009000003560200000000d60100000000d601000026060f00a203574d464301000000000 00100effd0000000001000000800300000000000080030000010000006c00000000000000000 00000030000001200000000000000000000005d000000ba01000020454d4600000100800300 000c00000001000000000000000000000000000000560500000003000040010000b30000000 0000000000000000000000000e2040038bb0200460000002c00000020000000454d462b0140 01001c000000100000000210c0db01000000600000006000000046000000500100004401000 0454d462b224004000c000000000000001e4009000c00000000000000244001000c00000000 0000003040020010000000040000000000803f214007000c00000000000000084000059c000 000900000000210c0db01000000000000000000000000000000000000000100000089504e47 0d0a1a0a0000000d494844520000000400000013080200000038403b0a00000001735247420 0aece1ce9000000097048597300000ec400000ec401952b0e1b0000001649444154185763fcff ff3f030c30c15940c6b0e60000ef8b032375825f520000000049454e44ae42608200000008400 10824000000180000000210c0db01000000030000000000000000000000000000001b400000 40000000340000000100000002000000000000bf000000bf000080400000984103000000000 080b3000080b3ffff7f40000080b3000080b3ffff97412100000008000000620000000c0000000 1000000150000000c00000004000000150000000c0000000400000051000000cc0000000000 000000000000030000001200000000000000000000000000000000000000040000001300000 05000000030000000800000004c000000000000002000cc0004000000130000002800000004 000000130000000100010000000000000000000000000000000000020000000000000000000 000ffffff00f0010101f0010101f0010101f0010101f0010101f0010101f0010101f0010101f0010 101f0010101f0010101f0010101f0010101f0010101f0010101f0010101f0010101f0010101f00 101014c00000064000000000000000000000003000000120000000000000000000000040000

{

}

∝

1. χ : ( − 1) .n n

n =1

= {-1, 2, -3, 4,-5, …..} adalah bukan barisan monoton, maka

salah satu bagian dari χ, yaitu: k : { − 1,−3,−5,−7,....} merupakan barisan monoton turun

2.

χ : {( − 2 )

}

n

+3

∝ n =1

= {1, 7, -5, 19,-29, …..} adalah bukan barisan monoton, maka

salah satu bagian dari χ, yaitu: k :{1, 7, 19,…..} merupakan barisan monoton naik

{

}

2 3. χ : n + 4

∝ n =1

= {5, 8, 13, 20, …..} adalah barisan monoton naik, maka salah

satu bagian dari χ, yaitu:

{ } k : n + 2 n =1 = {3,4,5,6,….}merupakan barisan monoton naik ∞

1   4. χ :  2n 

∝ n =1

1 1 1 1  , , ,  =  2 4 6 8  adalah barisan monoton turun, maka salah satu

bagian dari χ, yaitu: ∞

 1   2 k :  4n  n =1 =

1 1 1 1   , , ,   4 16 36 64  merupakan barisan bagian monoton turun

5.

χ : {( − 1) .5} n

∝ n =1

= {-5, 5, -5, 5,-5, …..} adalah bukan barisan monoton, maka

salah satu bagian dari χ, yaitu:

{ } k : − 5 n =1 = { − 5,−5,−5,−5,....} merupakan barisan monoton tetap ∞

{} k : 5 n =1 = {5, 5, 5, 5,……}merupakan barisan monoton tetap ∞