Nilpotent matrix and trace

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ . Suppose that the matrix $A$ is nilpotent and that the matrix $B$ commutes with $A$ . What can be said about $\operatorname{Tr}(AB)$ ?

Show Hint:

The matrix

AB

is also nilpotent. Why? And remember (or prove) that

0

is the only eigenvalue for a nilpotent matrix.

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Computing

\lim\limits_{n\to+\infty}\sum_{p=n+1}^{kn} \frac1 p

for

k > 1

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