Explain thoroughly please . The series > Cnx" converges when x = -2 and diverges when x = 5. What can be said about the following series?n=0(1) E(II) Een(-6)"Cn 2"n=0n=0(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 > Cnx" converges when x = -2 an
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Description: Explain thoroughly please . The series > Cnx" converges when x = -2 and diverges when x = 5. What can be said about the following series?n=0(1) E(II) Een(-6)"Cn 2"n=0n=0(a) Both series diverge(b) Both series converge(c) I converges; II diverges(d) I

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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 49E

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Explain thoroughly please

. The series > Cnx" converges when x = -2 and diverges when x = 5. What can be said about the following series?
n=0
(1) E
(II) Een(-6)"
Cn 2"
n=0
n=0
(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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The series > Cnx" converges when x = -2 and diverges when x = 5. What can be said about the following series? n=0 00 (1) E Cn2" (II) en(-6)" n=0 n=0 (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