Provide an appropriate response.
-If the series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( x + 6 ) ^ { n }$ is integrated term by term, for what value(s) of $x$ (if any) does the new series converge and for which the given series does not converge?
A) x = -5
B) x = -7
C) none
D) x = -7, x = -5

Question

Quizplus · Accepted Answer

When $x = -5$, the original series becomes $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 1 ) ^ { n }$, which is the alternating harmonic series. This series converges by the alternating series test. When integrated, the new series becomes $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac{(x+6)^{n+1}}{n+1}$, which converges for $x = -5$ by the alternating series test for infinite series. For any other value of $x$, the original series does not converge by the ratio test, so the integrated series also does not converge. Therefore, the answer is only choice A) x = -5.

Provide an Appropriate Response ∑n=0∞(−1)n(x+6)n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( x + 6 ) ^ { n }∑n=0∞​(−1)n(x+6)n

Provide an Appropriate Response $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( x + 6 ) ^ { n }$