# Sum of harmonic numbers $H_{n+k}$

I'm trying to take that sum: $$\sum_{k=1}^n H_{n+k}$$ So I transformed this sum to such: $$\sum_{i=1}^n iH_{2n+1-i}$$, unfortunately i can't make this sum out :( Hope You can help me, Thanks for attention!

You could replace $$H_{n+k}$$ with its definition and then change the order of summation.
A little bit simpler way is to note that your sum is $$S_{2n}-S_n$$, where $$S_n=\sum_{k=1}^{n}H_k=\sum_{k=1}^{n}\sum_{j=1}^{k}\frac{1}{j}=\sum_{j=1}^{n}\sum_{k=j}^{n}\frac{1}{j}\\=\sum_{j=1}^{n}\frac{n+1-j}{j}=(n+1)H_n-n.$$