For what values of x does the series converge conditionally?
-$$\sum _ { n = 0 } ^ { \infty } ( x + 2 ) ^ { n }$$

Question

Quizplus · Accepted Answer

The series $\sum _ { n = 0 } ^ { \infty } ( x + 2 ) ^ { n }$ is a geometric series with a common ratio of $r = x + 2$. For a geometric series to converge, the common ratio must satisfy $|r| < 1$. In this case, $|x + 2| < 1$, which simplifies to $-1 < x + 2 < 1$, or $-3 < x < -1$. Therefore, the series converges absolutely for $-3 < x < -1$, and there are no values of $x$ for which the series converges conditionally. Conditional convergence occurs when a series converges but does not converge absolutely, which is not applicable here.

For What Values of X Does the Series Converge Conditionally ∑n=0∞(x+2)n\sum _ { n = 0 } ^ { \infty } ( x + 2 ) ^ { n }n=0∑∞​(x+2)n

For What Values of X Does the Series Converge Conditionally $\sum _ { n = 0 } ^ { \infty } ( x + 2 ) ^ { n }$