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Catalan's constant

Value of the Dirichlet beta function with two as the argument

In mathematics, Catalan's constant G is the alternating sum of the reciprocals of the odd square numbers: G = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 − ⋯ . {\displaystyle G=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots .} Its numerical value is approximately (sequence A006752 in the OEIS) G = 0.915965594177219015054603514932384110774..., and it is also equal to β(2), where β is the Dirichlet beta function...

From Wikipedia

In mathematics, Catalan's constant G is the alternating sum of the reciprocals of the odd square numbers: G = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 − ⋯ . {\displaystyle G=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots .} Its numerical value is approximately (sequence A006752 in the OEIS) G = 0.915965594177219015054603514932384110774..., and it is also equal to β(2), where β is the Dirichlet beta function. Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.

Text: Wikipédia, CC BY-SA 4.0. · Image: Meyeds (CC BY-SA 4.0) ·

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