Explain thoroughly please The series $\sum_{n=0}^{\infty} c_n x^n$ converges when $x = -2$ and diverges when $x = 5$. What can be said about the following series?(I) $\sum_{n=0}^{\infty} c_n 2^n$(II) $\sum_{n=0}^{\infty} c_n (-6)^n$Options:(a) Both series diverge (b) Both series converge (c) I converges; II diverges (d) I cannot be determined; II diverges (e) Neither can be determined

Name: Explain thoroughly please The series $\sum_{n=0}^{\infty} c_n x^n$ c
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Description: Explain thoroughly please The series $\sum_{n=0}^{\infty} c_n x^n$ converges when $x = -2$ and diverges when $x = 5$. What can be said about the following series?(I) $\sum_{n=0}^{\infty} c_n 2^n$(II) $\sum_{n=0}^{\infty} c_n (-6)^n$Options:

Answered: The series > Cnx" converges when x = -2…

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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Explain thoroughly please

The series $\sum_{n=0}^{\infty} c_n x^n$ converges when $x = -2$ and diverges when $x = 5$. What can be said about the following series?

(I) $\sum_{n=0}^{\infty} c_n 2^n$

(II) $\sum_{n=0}^{\infty} c_n (-6)^n$

Options:
(a) Both series diverge
(b) Both series converge
(c) I converges; II diverges
(d) I cannot be determined; II diverges
(e) Neither can be determined

Expert Solution

Step 1

Here, we shall use the comparison test to conclude our solution.

The Limit Comparison Test for Series is given as

Let b(n) be a second series. Require that all a[n] and b[n] are positive.

If the $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} > 0$ , then ${\sum_{n = 0}^{\infty}}_{} a_{n}$ converges if and only if $\sum_{n = 0}^{\infty} b_{n}$ converges.
If the $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ , and $\sum_{n = 0}^{\infty} b_{n}$ converges, then ${\sum_{n = 0}^{\infty}}_{} a_{n}$ also converges.
If the $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \infty$ , and $\sum_{n = 0}^{\infty} b_{n}$ diverges, then ${\sum_{n = 0}^{\infty}}_{} a_{n}$ also diverges.

Step 2

II) We have to test the convergence of $\sum_{n = 0}^{\infty} c_{n} {(- 6)}^{n}$ .

Let $\sum_{n = 0}^{\infty} a_{n} = \sum_{n = 0}^{\infty} c_{n} {(- 6)}^{n}$ and $\sum_{n = 0}^{\infty} b_{n} = \sum_{n = 0}^{\infty} c_{n} {(- 2)}^{n}$ , where we know that $\sum_{n = 0}^{\infty} b_{n}$ converges.

With $a_{n} = c_{n} {(- 6)}^{n}, b_{n} = c_{n} {(- 2)}^{n}$ , we evaluate $\lim_{n \to \infty} \frac{a_{n}}{b_{n}}$ . Thus,
$\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = \lim_{n \to \infty} \frac{c_{n} {(- 6)}^{n}}{c_{n} {(- 2)}^{n}} = \lim_{n \to \infty} \frac{{(- 6)}^{n}}{{(- 2)}^{n}} = \lim_{n \to \infty} (\frac{n! . (- 6)}{n! . (- 2)}) (L' H o s p i t a l s' R u l e) = 3 > 0$

From Limit Comparison Test, we can see that $\sum_{n = 0}^{\infty} a_{n} = \sum_{n = 0}^{\infty} c_{n} {(- 6)}^{n}$ converges.

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