# Thread: Prove that harmonic series diverge

1. ## Prove that harmonic series diverge

Hi

Problem: Consider $\sum_{k=1}^{\infty} \frac{1}{k}$ . With $S_{n}=\sum_{k=1}^{n} \frac{1}{k}$ , prove that $S_{2^{n}}\geq 1+\frac{n}{2}$ by writing $S_{2^{n}}=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1} {4}\right) + ... + \left(\frac{1}{2^{n-1}+1}+...+\frac{1}{2^{n}}\right)$

Need some guidelines.

Thx!

2. Originally Posted by Twig
Hi

Problem: Consider $\sum_{k=1}^{\infty} \frac{1}{k}$ . With $S_{n}=\sum_{k=1}^{n} \frac{1}{k}$ , prove that $S_{2^{n}}\geq 1+\frac{n}{2}$ by writing $S_{2^{n}}=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1} {4}\right) + ... + \left(\frac{1}{2^{n-1}+1}+...+\frac{1}{2^{n}}\right)$

Need some guidelines.

Thx!