Summation: $\sum_{k=0}^{k=m}\frac{1}{k+1}$

Question

While working on a larger problem I reach a step where I need to know $\sum_{k=0}^{k=m}\frac{1}{k+1}$, as well as $\sum_0^n \frac{1}{k+2}$.

Is there a known formula/rule for these? (I am hoping there may be something similar to the known summations of infinite series).

Answer 1

You basically have: $$\sum_{k=0}^n\frac{1}{k+1}=\sum_{k=1}^{n+1}\frac{1}{k}=H(n+1)$$Where $H(n)$ is known as the $n$th harmonic number. The other sum you have is $H(n+2)-1$. Now the thing about the harmonic numbers is that there is no closed form formula for them. However, we do know that $$H(n)=\int_0^1\frac{1-x^n}{1-x}dx$$

Answer 2

Harmonic numbers can be written in terms of the digamma function $$ \sum_{k=0}^m\frac{1}{k+1} = \psi(m+2)+\gamma $$ But I doubt that is of use in the OP.