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I need to calculate the summation:

$$ \sum_{k=0}^N {N\choose{k}}\frac{x^k}{k^2} $$

I remember that this kind of summation should be calculated by integration and\or derivative, but I am not sure, and I don't know how to proceed. Any help is greatly appreciated!

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  • $\begingroup$ Sure about $k=0$? $\endgroup$
    – Did
    Nov 11, 2013 at 21:45

1 Answer 1

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If you differentiate with respect to $x$, $\frac{x^k}{k^2}$ becomes $\frac{x^{k-1}}{k}$.

Multiplying by $x$ and differentiating again, this becomes $x^{k-1}$.

Multiplying again by $x$, this becomes $x^k$ and you can do that summation.

Inverting the operations give an expression for the original sum.

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  • $\begingroup$ Thanks, so I think this summation does not have a closed form answer? $\endgroup$
    – Mah
    Nov 11, 2013 at 20:45

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