For what values of x does the series converge absolutely?
-$$\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }$$
A) $1 \leq x

Question

For what values of x does the series converge absolutely?
-$$\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }$$
A) $1 \leq x < 3$
B) $1 < x < 3$
C) $1 \leq x \leq 3$
D) $1 < x \leq 3$

Quizplus · Accepted Answer

The series $\sum _ { n = 0 } ^ { \infty } ( x - 2 ) ^ { n }$ is a geometric series with common ratio $r = x - 2$. For a geometric series to converge absolutely, the absolute value of the common ratio must be less than 1, i.e., $|x - 2| < 1$. Solving this inequality gives $1 < x < 3$.

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