A hyperbolic sum

Calculus Level 5

$\large \sum_{n=1}^\infty \dfrac{\coth(\pi n) }{n^3} = \dfrac{A\pi^B}C$

The equation above holds true for positive integers $A,B$ and $C$ with $A,C$ coprime. Find $A+B+C$ .

The answer is 190.

1 solution

Mark Hennings
Aug 5, 2016

The details of integrating $\frac{\pi \cot \pi z\, \coth \pi z}{z^3}$ about a suitable square contour are given in my solution here .

Actually I am unable to search any pdf or paper having entire complex analysis(advanced) such as which includes binomial representations and sums. If you have one, can you please share it?

Aditya Narayan Sharma - 4 years, 10 months ago

Here's an alternative solution .

Pi Han Goh - 4 years, 10 months ago

Many of the proofs in paper this require contour integration to complete. It is an alternative proof, but one using much more complex contour integration.

Mark Hennings - 4 years, 10 months ago

@Mark Hennings – Agreed. And here I am still trying to figure out a "non-contour integration" solution. I think that's impossible though....

Pi Han Goh - 4 years, 10 months ago

@Pi Han Goh – It's very much possible.

Ishan Singh - 4 years, 7 months ago

@Ishan Singh – You can't just write that comment without providing any evidence =P

Pi Han Goh - 4 years, 7 months ago

A hyperbolic sum

The answer is 190.

1 solution

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