# Convergence of a series based on reciprocals of prime factors of 2 & 3

1. Oct 18, 2004

### Kenshin

i dont even know where to start and i hate series. if someone could get me stared that would be great help. thanks

The terms of this series are reciprocals of positive integers whose only prime factors are 2s and 3s:

1+1/2+1/3+1/4+1/6+1/8+1/9+1/12+......

Show that this series converges and find its sum.

2. Oct 19, 2004

### arildno

Think of the infinite series as a sum of the folllowing subseries:
1. The subseries lacking "2" as a factor
2. The subseries with "2" as a single factor.
3. The subseries with 2 as a double factor (i.e, 2^2)
And so on..
We have the following sums:
1: 3/2
2: 1/2*3/2
3:1/4*3/2
and so on.
Hence, the total sum is 3.

Note that your infinite series is simply the Cauchy-product:
$$\sum_{n=0}^{\infty}\frac{1}{2^{n}}\sum_{m=0}^{\infty}\frac{1}{3^{m}}$$

End note:
Don't double post.

Last edited: Oct 19, 2004