How to find $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{12}+\cdots$?

Question

Find the sum of series $$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{12}+\cdots,$$ where the terms are the reciprocals of positive integers whose only prime factors are two's and three's:

Answer 1

Hint: factor $$\sum_{i=0}^\infty \sum_{j=0}^\infty \frac{1}{2^i 3^j}$$ as the product of a sum over $i$ and a sum over $j$.

Answer 2

$$\sum_{k=0}^{\infty}\frac{1}{2^k}\sum_{k=0}^{\infty}\frac{1}{3^k}=\frac{1}{1-\frac{1}{2}}\cdot\frac{1}{1-\frac{1}{3}}=3$$