# What is the sum of the infinite geometric series sum_(n=1)^oo2^n/5^(n-1) ?

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