Question

Find the sum of the convergent series. $$ 3-1+\frac{1}{3}-\frac{1}{9}+\cdots $$

Short answer

The sum of the convergent geometric series \(3-1+1/3-1/9+...\) is 2.25.

Step-by-step solution

Identify the initial term and common ratio

The first term \(a\) of the series is 3, and the common ratio \(r\) can be found by dividing the second term by the first term or the third term by the second term. In this case, \(r\) is \(-1/3\).

Check for Convergence

Before finding the sum of the geometric series, it's important to check if the series is convergent. A geometric series \(a + ar + ar^2 + ar^3 + ... \) is convergent if -1 < r < 1, which is true for our series because \(-1 < -1/3 < 1\). So, this series is convergent.

Find the Sum of the Series

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(r\) can be found by the formula \(S = a / (1 - r)\). Substituting our values, we have \(S = 3 / (1 - (-1/3)) = 3 / (1 + 1/3) = 3 / (4/3) = 9/4 = 2.25\).