# Thread: Summing an Infinite Series

1. ## Summing an Infinite Series

Hello. I'm stuck on another proof. Can anyone help me with this problem?

Let S of N be the Nth partial sum of the harmonic series

$\sum_{n=1}^{\infty}\frac{1}{n}$

a) Verify the following inequality for n=1,2,3. Then prove it for general n.

$\frac{1}{2^{n-1} + 2}$ + $\frac{1}{2^{n-1} + 1}$ + $\frac{1}{2^{n-1} + 3}$ + ... + $\frac{1}{2^n} \leq \frac{1}{2}$

b) Prove that S diverges by showing that $S of N \leq 1+\frac{n}{2}$ for N= $2^{n}$

Hint: Break up Sn into n+1 sums of length 1,2,4,8..., as in the following:

S of $2^{3}$ = 1 + (1/2) + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8)

Thank you!