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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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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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 +
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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 + ∝
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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 + ∝
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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 + ∝
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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
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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
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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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0900000a0101010101010101010101010101ffffff010b0f08010101010101010101010101010 10101010101010101101108010b120113140a01010d0e01010101010101010101010101ffffff 0101020607010101010101010101010101010101010101010101020607010b0c04010206070 10d0e01010101010101010101010101ffffff0101010200030401010101010101010101010101 0101010105000607010101050000080900000a0101010101010101010101010101ffffff01010 101010101010101010101010101010101010101010101010101010101010101010101010101 01010101010101010101010101ffffff010101010101010101010101010101010101010101010 1010101010101010101010101010101010101010101010101010101010101ffffff0101010101 010101010101010101010101010101010101010101010101010101010101010101010101010 101010101010101010101ffffff0c00000040092900aa00000000000000130035000000000004 0000002701ffff030000000000
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 ∞