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As we know, there is an asymptotic series for the digamma function when $z>0$ is a real number. $$ \psi(z)=\ln z+\sum_{n=1}^{\infty}{\frac{B_n}{nz^n}} $$ $B_n$ is the first Bernoulli numbers.

How to make a proof?

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Do you know that other famous asymptotic series, by Stirling? Might there be a connection to this one? –  Gerald Edgar May 7 '12 at 13:39
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up vote 1 down vote accepted

We can prove this, using Euler-Maclaurin Formula. Here is a introduction from Wikipedia. http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula

This is a quite easy problem. To Admin, You may be able to consider deleting this question, thanks. ^_^

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