Sum :$1+ 1/2 +1/3 +1/4 + 1/6 +1/8 \cdots $ , terms are reciprocals of the positive integers whose prime factors are $2$'s and $3$'s.

Question

I tried to break the series into different parts by two different methods

1. In terms of even and odd (no specific series is obtained in even, only Geometric progression formula can be applied on odd series)

2. In terms that contain powers of $2$ only, terms with $3$ to $1$st power, $3$ to $2$nd power,.. and so on

More specific pattern is obtained in the second method but I can't solve it further.

Help me further with the second method or any other one.

Any method given is appreciated.

Answer 1

Hint: $$ \sum_{x,y} \frac{ 1}{2^x 3^y } = \sum_x \frac{1}{2^x} \times \sum_y \frac{1}{3^y}.$$

Can you take it from here?