How to evalute <munderover> &#x2211;<!-- ∑ --> <mrow class="MJX-TeXAtom-ORD"> n

civilnogwu

civilnogwu

Answered question

2022-07-07

How to evalute n = 1 a n 2 + a 2 ?

Answer & Explanation

Jaruckigh

Jaruckigh

Beginner2022-07-08Added 11 answers

Firstly, we have the Fourier series for the hyperbolic cotangent:
coth x = 1 x + 2 n = 1 x x 2 + π 2 n 2
At x = a π in particular, with a little shuffling and algebra,
coth ( a π ) = 1 a π + 2 π n = 1 a a 2 + n 2
so
n = 1 a a 2 + n 2 = π 2 coth ( a π ) 1 2 a

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