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## Homework Statement

## Homework Equations

Down

## The Attempt at a Solution

As you see in the solution, I am confused as to why the sum of residues is required.

My question is the sum:

$$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$

**Question #1:**

-Why is the beginning n=1 the residue is for

**all**$$n \in (-\infty, \infty)$$ so the beginning of the sum should be $$n=-\infty$$ right?

**Question #2:**

Also in the residue theorem, it states:

$$\oint_{C} f(z) dz = (2\pi i)(\sum \text{Res})$$

Where is the 2pi(i) gone in the solution of theirs?

Thanks!