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find a closed form for the sum of the zeta function $\zeta(k)$ for $k$ runs from $1$ to $n$. I need this to find the sum of an infinite series involving the zeta function at the natural numbers. Any help is nice.

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up vote 2 down vote accepted

Since you aren't very specific with what you're asking, I only have a vague idea of what you might be looking for so I will give you a couple things. I think what you're asking for is the following:

Faulhaber's Formula: $$ \sum_{k=1}^n k^x=\frac{1}{x+1}\sum_{k=0}^x \binom{x+1}{k}B_kn^{x+1-k} $$ where $B_k$ are the Bernoulli numbers. Of course, these provide good approximations when the powers are in the integers. Generally, you'd need Euler-Maclaurin. These sums also have expressions using generalized harmonic numbers.

Depending on the context in which you are working, you may find one or more of the following useful:

$$ \sum_{n=2}^\infty \zeta(n)-1=1 $$ $$ \sum_{n=1}^\infty \zeta(2n+1)-1=\frac{1}{4} $$ $$ \sum_{n=1}^\infty \zeta(2n)-1=\frac{3}{4} $$

And of course, I'm assuming you know the Hurwitz Zeta function: $$ \zeta(s,q)=\sum_{n=0}^\infty \frac{1}{(q+n)^s} $$

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