Tutesheet 1 Phool Singh

Form No. ITM‐FRM‐66(REV.00)  Institute of Technology and  

Management Department: Applied Sciences and Humanites

Tutorial Sheet No. 01

Subject Name : Mathematics I Subject Code : ASL 101 Unit/Title: Infinite Series

Issue Date:

Pages : 02

Faculty :

Semester:I

Phool Singh

1. 2. 3.

1 2 3 + ………………∞ (divergent ) + + √1+√2 √2 + √3 √3 + √4 1 + 1 + 1 + ……………. ∞ (Convergent a, b≠0) a.12+b a.22+b a.32+b 1 ∑ (Convergent if α>1 and divergent if α ≤1)

n 4. 5

∑

α +β n

(√n3 + 1 - √n3 )

Ans. Convergent

Check the convergence of the following series: 1 ⎛1⎞ tan ⎜ ⎟ (Convergent) ∑ ⎝n⎠ n n =1 1 + 1 + 1 +………..∞ (Convergent) , a ≠ 0 2 2 2 (x-a) (x –2a) (x-3a) ∑ 1 x>0 (Convergent for all values of x except 1) xn + x-n x +2x2 + 3x3 + 4x4 + ………∞ ,x>0 Ans. : Cgt for x < 1 and dgt for x ≥ 1 ∑ n2 (Convergent) n ∑ n2 + a (Convergent) n 2 +a ∑ xn for x>0 (Convergent for x ≤ 1, divergent for x >1) 2n(2n-1) 1 + a + a(a+1) + a (a+1) (a+2) + …….∞ (convergent for a ≤ 0; divergent if a >0) 1.2 1.2 .3 2 2 x + 2.4 x + 2.4.6 x3 +..…… ∞ (x>0) (Cgt for x<3/2 ; Dgt for x≥3/2) 5 5.8 5.8.11 3 x + 1 x + 1.3 x5 + 1.3.5 x7 +……..…∞ x>0 (x>1 Divergt. x ≤ 1 Conv) 1 2 3 2.4 5 2.4.6 7 ∞

6. 7. 8. 9. 10. 11. 12. 13. 14.

15.

a + x + (a+2x)2 + (a + 3x)3 +….. ∞ 1! 2! 3! (Convergent for x<1/e and divergent for x ≥ 1/e)

Form No. ITM‐FRM‐66(REV.00)   

16.

Test the convergence of the series (2004) 1 1 1 Convergent + + + ...... (a) 1.2.3 2.3.4 3.4.5 n 2 (n + 1) 2 Convergent (b) ∑ n! 2 2 x 2 33 x 3 4 4 x 4 x < 1/e Convergent x ≥ 1/e Divergent + + + ..... (c) x + 2! 3! 4! 17. (a) Test the convergence or divergence of the series ∞

∑ ⎡⎢⎣ 1

(n 2 + 1) − n⎤ ⎥⎦

divergent

(b) Test the following series for convergence 1 x2 1 3 5 x4 1 3 5 7 9 x2 + . . . + . . . . . + ---------1+ . 2 4 2 4 6 8 2 4 6 8 10 12

(2006)

Convergent for x 2 ≤ 1, divergent for x 2 >1