What is the sum of the infinite geometric series $\infty \sum n = 1 \frac{2^{n}}{5^{n - 1}}$ $sum_(n=1)^oo2^n/5^(n-1)$ ?

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What is the sum of the infinite geometric series $\infty \sum n = 1 \frac{2^{n}}{5^{n - 1}}$ $sum_(n=1)^oo2^n/5^(n-1)$ ?

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Answer 1 · 2014-07-23T21:47:19

This series can be rewritten as $\infty \sum n = 1 2 {(\frac{2}{5})}^{n - 1}$ $sum_{n=1}^\infty 2(2/5)^{n-1}$ and is thus a geometric series with first term 2 and common ratio $\frac{2}{5}$ $2/5$ . Since the common ratio is less than 1 (in absolute value), the series is convergent and its sum is given by $\frac{first term}{1 - common ratio} = \frac{2}{1 - \frac{2}{5}} = \frac{10}{3}$ $"first term"/(1-"common ratio")=2/(1-2/5)=10/3$ . See Geometric Series

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What is the sum of the infinite geometric series $\infty \sum n = 1 \frac{2^{n}}{5^{n - 1}}$ $sum_(n=1)^oo2^n/5^(n-1)$ ?

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