# Harmonic series proof

1. Oct 5, 2009

### James889

Hai,

The harmonic series is given by: $$H_{n} = \sum_{i=1}^n \frac{1}{i}$$

I need to prove that for all positive integers:
$$\sum_{j=1}^n H_{j} = (n+1)H_{n} -n$$

So i have
$$H_{5} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} = \frac{137}{60}$$

$$H_{5} \neq (5+1)*\frac{137}{60} -5$$

Have i missed something here?

Please excuse my epic fail math skills...

Last edited: Oct 5, 2009
2. Oct 5, 2009

### Office_Shredder

Staff Emeritus
So for your H5 example what they want you to sum is

H1 + H2 + H3 + H4 + H5

When I did that I got what the problem tells you you will get