An Interesting Infinite Sum of Products

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INFINITE SERIES:

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[Blaqjaqshellaq]

The Taylor series for Arctan(1) is 1 - 1/3 + 1/5 - 1/7 +..., which equals pi/4, so half of this series is pi/8.

[itsphoenixingtime]

This is a rephrasing of the Madhava–Leibniz [and Gregory?] series. Expanding partial fractions and writing out the sum in full gives the familiar series.

[VincentKok]

Wolfram alpha can do it you just need to specify the problem better, if you input:
sum_{n=0}^{inf} 1/((4n + 1)(4n+3))
It gives you pi/8

[inigom]

You can also say: S = (1/2) * arctan 1 = (1/2) * pi/4

[sadekmebarka]

Appliquer la formule: 1/(a×b)=1/(a-b)×(1/b-1/a)