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T-n-0-1-4n-1-2-1-4n-3-2-




Question Number 212360 by MrGaster last updated on 11/Oct/24
       T=Σ_(n=0) ^∞ ((1/((4n+1)^2 ))−(1/((4n+3)^2 ))
$$ \\ $$$$\:\:\:\:\:{T}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{\mathrm{1}}{\left(\mathrm{4}{n}+\mathrm{1}\right)^{\mathrm{2}} }−\frac{\mathrm{1}}{\left(\mathrm{4}{n}+\mathrm{3}\right)^{\mathrm{2}} }\right. \\ $$
Answered by Berbere last updated on 15/Oct/24
=Σ_(n≥0) (((−1)^n )/((2n+1)^2 ))=C catalane constante
$$=\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }={C}\:{catalane}\:{constante} \\ $$

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