If \sum_{i=1}^\infty\tan^{-1}(\frac{1}{2i^2})=t

Casertowm7

Casertowm7

Answered question

2022-02-26

If
i=1tan1(12i2)=t

Answer & Explanation

Lucian Robin

Lucian Robin

Beginner2022-02-27Added 4 answers

So:
12k2=12k112k+11+12k112k+1
and the identity
tan(xy)=tan(x)tan(y)1+tan(x)tan(y)
we get
k=1tan1(12k2)=k=1tan1(12k1)tan1(12k+1)
=tan1(1)
=π4

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