Question

Determine the convergence or divergence of the series. $$ \sum_{n=0}^{\infty} \frac{1}{4^{n}} $$

Short answer

The series \( \sum_{n=0}^{\infty} \frac{1}{4^{n}} \) converges and its sum is \( S = \frac{4}{3} \).

Step-by-step solution

Identify the Form

Recognize that the series \( \sum_{n=0}^{\infty} \frac{1}{4^{n}} \) is a geometric series, where \( a = 1 \) (the first term when \( n=0 \)) and the common ratio \( r = \frac{1}{4} \).

Applying the Convergence Test for Geometric Series

The geometric series converges if and only if the absolute value of \( r \) is less than 1. Here, \( |r| = |\frac{1}{4}| = \frac{1}{4} < 1 \). Thus, this geometric series converges.

Find the Sum of the Convergent Series

For a convergent geometric series, the sum can be found using the formula \( S = \frac{a}{1-r} \). Substituting the values of \( a \) and \( r \) into the formula, the sum becomes \( S = \frac{1}{1 - \frac{1}{4}} = \frac{4}{3} \).