Question

determine whether the infinite geometric series converges or diverges. if it converges, find its sum.
\\( \sum_{k=1}^{\infty} \frac{1}{6}5^{k - 1} \\)

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

Explanation:

Step1: Identify first term and ratio

The series is $\sum_{k=1}^{\infty} \frac{1}{6}5^{k-1}$. For $k=1$, first term $a=\frac{1}{6}5^{0}=\frac{1}{6}$. Common ratio $r=5$.

Step2: Check convergence condition

An infinite geometric series converges if $|r|<1$. Here $|5|=5>1$, so it diverges.

Answer:

B. The series diverges.