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I have the following sum:

$$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$

Are there any methods that I can use to speed up the convergence of the sequence generated by taking the partial sums? I have not found anything on-line, but I also don't know much about how to do this type of transformation while keeping the limit of the sequence the same.

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  • $\begingroup$ en.wikipedia.org/wiki/Series_acceleration is devoted to such tricks. It is also interesting to recall, from Apery's work, that: $$\zeta(3)=\frac{5}{2}\sum_{n\geq 1}\frac{(-1)^{n+1}}{n^3\binom{2n}{n}}.$$ $\endgroup$
    – Jack D'Aurizio
    Commented Mar 2, 2015 at 15:54

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Well, there is the Euler-Maclaurin Formula. Not guaranteed to work, but it may be worth a shot.

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