Question

determine whether the infinite geometric series converges or diverges. if it converges, find its sum.
$1+\frac{1}{6}+\frac{1}{36}+\dots$

select the correct choice below and fill in any answer boxes within your choice.
\\(\bigcirc\\) a. the series converges. the sum of the series is \\(\square\\).
(type an integer or a simplified fraction.)
\\(\bigcirc\\) b. the series diverges.

Explanation:

Step1: Identify first term and ratio

First term $a = 1$. Ratio $r = \frac{1/6}{1} = \frac{1}{6}$.

Step2: Check convergence condition

$|r| = \frac{1}{6} < 1$, so series converges.

Step3: Calculate sum of series

Use formula $S = \frac{a}{1 - r}$. Substitute values: $S = \frac{1}{1 - 1/6} = \frac{1}{5/6} = \frac{6}{5}$.

Answer:

A. The series converges. The sum of the series is $\frac{6}{5}$.