= Solution
The <hyperbolic cotangent> is
$$
\coth x=\frac{e^x+e^{-x}}{e^x-e^{-x}}=\frac{e^{2x}+1}{e^{2x}-1},\qquad x\ne0.
$$
Expand the numerator and denominator using the exponential <Maclaurin series>:
$$
\coth x=\frac1x\frac{1+x^2/2+x^4/24+O(x^6)}{1+x^2/6+x^4/120+O(x^6)}
=\boxed{\frac1x+\frac x3-\frac{x^3}{45}+O(x^5).}
$$
This is a <Laurent series>, rather than an ordinary <Maclaurin series>: $\coth x$ has a simple <pole> at zero. The printed request should be understood as the expansion obtained from the exponential series, including this unavoidable $x^{-1}$ term.
Back to article page
