# Math Help - Express as power series

1. ## Express as power series

Is it possible to express $\frac{1}{e^t+e^{-t}}$ as a power series in $e^{-t}$?

So can we find $a_n$ with $\frac{1}{e^t+e^{-t}} = \sum_{n=0}^\infty a_n (e^{-t})^n$ ?

2. Setting $e^{-t} = x$ we obtain...

$\frac{1}{e^{t} + e^{-t}} = \frac{1}{x+\frac{1}{x}} = \frac{x}{1+x^{2}} =$

$= x\cdot (1-x^{2} + x^{4} - \dots) = x - x^{3} + x^{5} - \dots$ (1)

... so that is...

$\frac{1}{e^{t} + e^{-t}} = e^{-t} - e^{-3t} + e^{-5t} - \dots$ (2)

Because the series (1) converges for $|x|<1$, the series (2) converges for $t>0$ ...

Kind regards

$\chi$ $\sigma$

3. Didn't we just have this exact problem recently?