Catalan's Constant - Quickly Converging Series

Quickly Converging Series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:

\begin{align} G & = 3 \sum_{n=0}^\infty \frac{1}{2^{4n}} \left( -\frac{1}{2(8n+2)^2} +\frac{1}{2^2(8n+3)^2} -\frac{1}{2^3(8n+5)^2} +\frac{1}{2^3(8n+6)^2} -\frac{1}{2^4(8n+7)^2} +\frac{1}{2(8n+1)^2} \right) \\ & {}\quad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}} \left( \frac{1}{2^4(8n+2)^2} +\frac{1}{2^6(8n+3)^2} -\frac{1}{2^9(8n+5)^2} -\frac{1}{2^{10} (8n+6)^2} -\frac{1}{2^{12} (8n+7)^2} +\frac{1}{2^3(8n+1)^2} \right) \end{align}

and

The theoretical foundations for such series is given by Broadhurst (the first formula) and Ramanujan (the second formula). The algorithms for fast evaluation of the Catalan constant is constructed by E. Karatsuba.

Read more about this topic:  Catalan's Constant

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