Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } e ^ { - 10 n }$$
A) Converges; $\frac { e ^ { - 10 } } { e ^ { - 10 } - 1 }$
B) Converges; $\frac { e ^ { 10 } } { e ^ { 10 } - 1 }$
C) Converges; $\frac { 1 } { e ^ { - 10 } - 1 }$
D) Diverges

Question

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = e^0 = 1$ and common ratio $r = e^{-10}$. A geometric series converges if $|r| < 1$, which is true in this case since $e^{-10} > 0$ but less than 1. The sum of a converging geometric series is given by $S = \frac{a}{1 - r}$, so substituting $a = 1$ and $r = e^{-10}$, we get $S = \frac{1}{1 - e^{-10}} = \frac{e^{10}}{e^{10} - 1}$, which matches option B.

Determine If the Series Converges or Diverges; If the Series ∑n=0∞e−10n\sum _ { n = 0 } ^ { \infty } e ^ { - 10 n }n=0∑∞​e−10n

Determine If the Series Converges or Diverges; If the Series $\sum _ { n = 0 } ^ { \infty } e ^ { - 10 n }$