What is the result of sum_(i=1)^infty 1/(2i-1)^2

NormmodulxEE

NormmodulxEE

Answered question

2022-11-23

What is the result of
i = 1 1 ( 2 i 1 ) 2

Answer & Explanation

Zaria Tanner

Zaria Tanner

Beginner2022-11-24Added 5 answers

First, assuming that you know the identity,
S = 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + = π 2 6
Then,
1 2 2 + 1 4 2 + 1 6 2 + 1 8 2 = 1 2 2 ( 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ) = S 4
Therefore,
1 2 + 1 3 2 + 1 5 2 + 1 7 2 + = S S 4 = 3 S 4 = π 2 8
We could do this re-arrangement in last step since series convergent absolutely(trivial).

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