I'm in trouble with this homework.
Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive integers whose only prime factors are 2 and 3.
Hint: write the series as a product of two geometric series.
Ok, I found some patterns in these numbers, but I can't find the two series.
Best Answer
Hint: $$ = \left( 1+\frac{1}2 + \frac{1}4 + \cdots \right) \left( 1+ \frac{1}3 + \frac{1}9 + \cdots \right)$$.