Fast converging series for some famous constants

Details Fast converging series for some famous constants Fast converging series for some famous constants

2010-10-20

arxiv.org/abs/1010.4298v4

Mathematics

Number Theory

10 pages. Refined version

Scientific paper

For k=0,1,2,... let $H_k^{(2)}:=\sum_{j=1}^k 1/j^2$ and $\bar H^{(2)}_k:=\sum_{j=1}^k 1/(2j-1)^2$. In this paper we give a new type of exponentially converging series (involving $H_k^{(2)}$ or $\bar H^{(2)}_k$) for certain famous constants. For example, we show that $$\sum_{k=1}^\infty \binom(2k,k)H_{k-1}^{(2)}/k^2=\pi^4/1944, \sum_{k=1}^\infty\binom(2k,k)\bar H^{(2)}_k/((2k+1)(-16)^k)=-(log^3\phi)/3, \sum_{k=1}^\infty L_{2k}H_{k-1}^{(2)}/(k^2\binom(2k,k))=41\pi^4/7500, \sum_{k=1}^\infty\binom(2k,k)L_{2k+1}\bar H^{(2)}_k/((2k+1)16^k)=13\pi^3/1500,$$ where $\phi$ is the golden ratio (\sqrt 5+1)/2, and L_0,L_1,L_2,... are Lucas numbers given by L_0=2, L_1=1, and L_{n+1}=L_n+L_{n-1} (n=1,2,3,...).

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