Question

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

Answer 1

This series can be rewritten as `sum_{n=1}^\infty 2(2/5)^{n-1}` and is thus a geometric series with first term 2 and common ratio `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