Answered: 6. Are the following series convergent?… | bartleby

Advanced Engineering Mathematics

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

6. Are the following series convergent? Why?
(a).
(b)
8
n=1
2
(3n - 1)1.3
n=1
n² - 1
2n² + n

Expert Solution

Step 1

The given series are ,

(a). $\sum_{n = 1}^{\infty} \frac{2}{{(3 n - 1)}^{1.3}}$

(b). $\sum_{n = 1}^{\infty} \frac{n^{2} - 1}{2 n^{2} + n}$

$(.) N e c e s s a r y c o n d i t i o n f o r c o n v e r g e n c e :$ If the series $\sum_{n = 1}^{\infty} a_{n}$ converges , then $\lim_{n \to \infty} a_{n} = 0$ .

$(.) C o n t r a p o s i t i v e o f a b o v e s t a t e m e n t :$ If $\lim_{n \to \infty} a_{n} \neq 0$ then series $\sum_{n = 1}^{\infty} a_{n}$ does not converge .

$(.) L i m i t c o m p a r i s o n t e s t :$ If $\sum_{n = 1}^{\infty} a_{n}$ and $\sum_{n = 1}^{\infty} b_{n}$ are two positive term series such that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = l$ ; where $' l'$ is a finite and non - zero real number , then both series $\sum_{n = 1}^{\infty} a_{n} & \sum_{n = 1}^{\infty} b_{n}$ converge or diverge together .

$(.) p - s e r i e s t e s t :$ The series $\sum_{n = 1}^{\infty} \frac{1}{n^{p}}$ is convergent if $p > 1$ and divergent if $p \leq 1$ .

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(c) Find the sum of each of the following series. (i) E n(n – 1)x", |x|< 1 n=2 n² – n (ii) 2 2" (iii) E 2" n=2 n=1 (?
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