What is the name for a series that uses exponential functions of a variable, rather than powers of that variable, to approximate a function? Consider the function $\text{sech}(\pi \frac{x}{2})$ and suppose that we wish to find an approximation for this function at large $x$. One route seems to be to write 
$$ \text{sech}(\pi \frac{x}{2}) = \frac{2}{\exp{ \frac{\pi x}{2}} + \exp{\frac{-\pi x}{2}}}, $$ 
then to formally let $\exp{ \frac{\pi x}{2}} = y$ and find the power series of 
$$ \frac{2}{y+\frac{1}{y}} \approx 2(\frac{1}{y} - \frac{1}{y^3}) + \mathcal{O}(y^{-5}),  \ \  y \rightarrow  +\infty.$$
If we substitute for the original expression we have 
$$ \text{sech}(\pi \frac{x}{2}) \approx 2( e^{-\frac{ \pi x}{2}} - e^{-\frac{3 \pi x}{2}}) + \mathcal{O}(e^{- \frac{5 \pi x}{2}}), \ \ x \rightarrow +\infty. $$
I have investigated the partial sums numerically and they seem to converge to the original function for $x>0$. However, this process seems dubious, because if you try the same trick to find the behavior at small $x$, you get a result that seems valid (numerically) ONLY at the origin.
1) Is there a name for this type of approximation?
2) Under what conditions is this type of expansion valid?
If the explanation is too detailed for a SE post, I would consider a pointer to a common reference an equally good answer. I apologize if this is a duplicate, but since I don't know the name of this expansion, I didn't know what to search for.
 A: Some groundwork.
The expansion
$$
\frac{2}{y + \frac{1}{y}} = \frac{2}{y} \frac{1}{1+\frac{1}{y^2}} = \frac{2}{y} \sum_{k=0}^{\infty} \frac{(-1)^k}{y^{2k}}
$$
holds if and only if $|1/y^2| < 1$, which is equivalent to $|y| > 1$.  So, returning to the original variables via $y = e^{\pi x/2},$ we see that
$$
\operatorname{sech}\left(\frac{\pi x}{2}\right) = 2 e^{-\pi x/2} \sum_{k=0}^{\infty} (-1)^k e^{-k \pi x}
$$
if and only if $\pi x/2 > 0$, which is equivalent to $x > 0$.
The Big O estimates.
For a power series $\sum a_k x^k$ with radius of convergence $R$, the $n^\text{th}$ tail of the series,
$$
\sum_{k=n+1}^\infty a_k x^k,
$$
is $O(x^{n+1})$ when $|x| \leq R-\epsilon$ for any fixed $\epsilon > 0$.  The constant implicit in the Big O will, in most cases, depend on the $\epsilon$ you choose.
The Big O estimate doesn't necessarily hold on the whole region $|x| < R$ since the series may diverge on the boundary $|z| = R$.  This is exactly what happens in the above case.  Indeed, it is only true that
$$
\operatorname{sech}\left(\frac{\pi x}{2}\right) = 2 e^{-\pi x/2} \left(\sum_{k=0}^{n} (-1)^k e^{-k \pi x} + O(e^{-(n+1)\pi x})\right)
$$
when $x \geq \epsilon$ for any fixed $\epsilon > 0$.
To get an estimate which holds for $x$ in a neighborhood of $0$ you'd be better off just using the Taylor series
$$
\operatorname{sech}\left(\frac{\pi x}{2}\right) = 1 - \frac{\pi^2}{8}x^2 + \frac{5 \pi^4}{384}x^4 + O(x^6),
$$
where the Big O holds for $|x| \leq 1-\epsilon$ for any fixed $\epsilon > 0$.
Any book on asymptotic analysis should expand on these ideas if you're interested in learning more about them. Approximations of this type are truncated asymptotic expansions.
