Determine the sum of the series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 2 } { e } \right) ^ { n }$ .
A) $\frac { 2 } { \mathrm { e } }$
B) $\frac { \mathrm { e } } { \mathrm { e } + 2 }$
C) 1 - $\frac { 2 } { \mathrm { e } }$
D) none of these

Question

Quizplus · Accepted Answer

The series is a geometric series with first term $a = 1$ and common ratio $r = -\frac{2}{e}$. The sum of an infinite geometric series is $\frac{a}{1-r}$ if $|r| < 1$. Here, $|-\frac{2}{e}| < 1$, so the sum is $\frac{1}{1 + \frac{2}{e}} = \frac{e}{e + 2}$.

Determine the Sum of the Series ∑n=0∞(−1)n(2e)n\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 2 } { e } \right) ^ { n }∑n=0∞​(−1)n(e2​)n

Determine the Sum of the Series $\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \left( \frac { 2 } { e } \right) ^ { n }$