# Thread: [SOLVED] Clarification on harmonic series

1. ## [SOLVED] Clarification on harmonic series

I know that $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$ diverges, but does $\displaystyle \sum_{n=k}^{\infty}\frac{1}{n}$ diverge for any $\displaystyle k > 1$? It's been a while since I've dealt with series.

2. Originally Posted by Pinkk
I know that $\displaystyle \sum_{n=1}^{\infty}\frac{1}{n}$ diverges, but does $\displaystyle \sum_{n=k}^{\infty}\frac{1}{n}$ diverge for any $\displaystyle k > 1$? It's been a while since I've dealt with series.
Yes, since $\displaystyle \sum_{n=k}^{\infty}\frac{1}{n}={\color{red}\sum_{n =1}^{\infty}\frac{1}{n}}-\sum_{n=1}^{k-1}\frac{1}{n}$ where we have the harmonic series appear again...

3. Ah okay, thanks. That jogs my memory a bit.

4. $\displaystyle \sum\limits_{n=1}^{\infty }{\frac{1}{n}}=1+\frac{1}{2}+\cdots +\sum\limits_{n\ge k}{\frac{1}{n}},$ since the series of the LHS diverges, then obviously $\displaystyle \sum_{n\ge k}\frac1n$ diverges.

(ohh, i'm a little slow today!)