Find the sum of the geometric series for those x for which the series converges.
-$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }$$
A) $\frac { 1 } { 1 - 64 x ^ { 2 } }$
B) $\frac { 64 } { 1 + 64 x ^ { 2 } }$
C) $\frac { 8 } { 1 + 64 x ^ { 2 } }$
D) $\frac { 1 } { 1 + 64 x ^ { 2 } }$

Question

Quizplus · Accepted Answer

The given series is a geometric series with first term $1$ and common ratio $-(8x)^2$. For the series to converge, we need $|-(8x)^2|<1$, which is equivalent to $|x|<\frac{1}{8}$. Thus, the sum of the series is $\frac{1}{1-(-(8x)^2)}=\boxed{	extbf{(D) }\frac{1}{1+64x^2}}$.

Find the Sum of the Geometric Series for Those X ∑n=0∞(−1)n(8x)2n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }n=0∑∞​(−1)n(8x)2n

Find the Sum of the Geometric Series for Those X $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } ( 8 x ) ^ { 2 n }$