Taylor Series of Hyperbolic Cotangent Coth(x)

Question

Expanding about 0 gets me a divergence on the first term, and the wikipedia article says nothing about how to derive it other than taylor series. It makes me think I'm supposed to use Laurent Series, it's been two years since I did that and I don't remember it. Is that the only way to find the expansion?

Answer 1

According to http://en.wikipedia.org/wiki/Hyperbolic_function#Taylor_series_expressions, the series form of $\coth x$ is $\dfrac{1}{x}+\sum\limits_{n=1}^\infty\dfrac{4^nB_{2n}x^{2n-1}}{(2n)!}$ , where $B_n$ is the $n$th Bernoulli number and only suitable for $0<|x|< \pi$ .