Question

Prove that $H_1+H_2+\dots +H_n=(n+1)H_n-n$

Short answer

$H_{2^{\mathrm{\prime }}}\le 1+n$is a non negative integer.

Step-by-step solution

Step: 1

If $H_1=1$

it is true for n=1.

Step: 2

Let P(k) be true.

$H_1+H_2+\dots +H_k=(k+1)H_k-k$

We need to prove that P(k+1) is true.

Step: 3

$\begin{array}{r}{H}_1+H_2+\dots +H_k\\ (k+1)H_k-k+H_{k+1}\\ =(k+1)H_k-k\end{array}$

It is true for P (k+1) is true