Determine if the series converges or diverges; if the series converges, find its sum.
-$$\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }$$
A) Converges; $\frac { 6 - \sqrt { 6 } } { 5 }$
B) Converges; $\frac { 6 + \sqrt { 6 } } { 5 }$
C) Converges; $\frac { \sqrt { 6 } - 6 } { 5 }$
D) Diverges

Question

Quizplus · Accepted Answer

This is a geometric series with first term $1$ and common ratio $\frac{1}{\sqrt{6}}$, which means it converges if and only if $|r|<1$ where $r$ is the common ratio. In this case, $\left| \frac{1}{\sqrt{6}}\right| <1$, so the series converges. 
To find its sum, we use the formula for the sum of an infinite geometric series:
$$\frac{a}{1-r}=\frac{1}{1-\frac{1}{\sqrt{6}}}=\frac{1}{1-\frac{\sqrt{6}}{6}}=\frac{6}{6-\sqrt{6}}=\frac{6+\sqrt{6}}{5}$$
Therefore, the answer is (B).

Determine If the Series Converges or Diverges; If the Series ∑n=0∞1(6)n\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }n=0∑∞​(6​)n1​

Determine If the Series Converges or Diverges; If the Series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { ( \sqrt { 6 } ) ^ { n } }$