Home -> Solved problems -> Find the infinite sum (1/3)^k

### Solution

First, let’s show that the series converges: $|\frac{1}{3}|=\frac{1}{3}<1$ Thus, the series is convergent.
\begin{aligned} &S=\sum_{k=1}^{\infty} \frac{1}{3^{k}}=\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots \\\\\\ &3 S=1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots \\\\\\ &3 S-S=1 \\\\\\ &S=\frac{1}{2} \end{aligned}
Home -> Solved problems -> Find the infinite sum (1/3)^k

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