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

Question

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = \left( \frac{\pi}{e} \right)^1 = \frac{\pi}{e}$ and common ratio $r = \frac{\pi}{e}$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, provided $|r| < 1$. Substituting $a = \frac{\pi}{e}$ and $r = \frac{\pi}{e}$, we get $S = \frac{\frac{\pi}{e}}{1 - \frac{\pi}{e}} = \frac{\pi}{e - \pi}$.

Determine the Sum of the Series ∑n=1∞(πe)n\sum _ { n = 1 } ^ { \infty } \left( \frac { \pi } { e } \right) ^ { n }∑n=1∞​(eπ​)n

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