Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
\[
\sum_{k=1}^{\infty} 4\left(-\frac{1}{3}\right)^{k-1}
\]
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.
Answer
Final Answer: The series converges. The sum of the series is \(\boxed{3}\).
Step 1 ：The given series is a geometric series with common ratio \(r = -\frac{1}{3}\) and first term \(a = 4\).
Step 2 ：A geometric series converges if and only if \(|r| < 1\). In this case, \(|r| = \frac{1}{3} < 1\), so the series converges.
Step 3 ：The sum of an infinite geometric series is given by the formula \(\frac{a}{1 - r}\).
Step 4 ：Substituting the values of \(a\) and \(r\) into the formula, we get the sum as 3.
Step 5 ：Final Answer: The series converges. The sum of the series is \(\boxed{3}\).
