Step 1: Understanding the Concept:
The given series is a geometric series. A geometric series has the form \(\sum_{n=0}^\infty ar^n\). We need to find the values of \(x\) for which this series converges.
Step 2: Key Formula or Approach:
A geometric series \(\sum_{n=0}^\infty ar^n\) converges if and only if the absolute value of the common ratio \(r\) is less than 1, i.e., \(|r| < 1\).
The series diverges if \(|r| \ge 1\).
For the given series, we need to identify the common ratio \(r\) and apply this condition.
Step 3: Detailed Explanation:
The given series is \(\sum_{n=0}^\infty (2x)^n\).
This is a geometric series with the first term \(a = (2x)^0 = 1\) and the common ratio \(r = 2x\).
For the series to converge, we must have \(|r| < 1\).
\[ |2x| < 1 \]
This inequality can be split into two parts:
\[ -1 < 2x < 1 \]
To solve for \(x\), we divide all parts of the inequality by 2:
\[ \frac{-1}{2} < x < \frac{1}{2} \]
This is the interval of convergence for the series.
Step 4: Final Answer:
The series converges if \(-\frac{1}{2} < x < \frac{1}{2}\), which corresponds to option (B).