Variation of the sum of 1 / (n +k) with k =1, to n

Question

Could you help me find the variation of the sum sequence $$S(n)=\sum^n_{k=1}(n+k).$$ I tried to do $S(n+1) - S(n)$ but I can't simplify. Any ideas?

Answer 1

Note: the title and question don't agree.

Assuming you mean $\sum_{k=1}^n(n+k)$, you have that$$\sum_{k=1}^n(n+k)=\frac12 (n+3n^2)$$and so$$S(n+1)-S(n)=1+2n.$$ If you meant $\sum_{k=1}^n1/(n+k)$ then$$\sum_{k=1}^n\frac{1}{n+k}=\psi(1+2n)-\psi(1+n)$$ where $\psi$ is the Digamma function. Hence,$$S(n+1)-S(n)=\psi(2+2n)-\psi(1+2n)=\frac{1}{1+2n}.$$