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