I have this eq. $ \frac{x}{x-1} + e^{(x-1)/\epsilon} - \epsilon = 0. $
I need to derive an asymptotic expansion for any bounded roots $x(\epsilon)$ this equation might have, to find the coefficient of the $n$th (‘general’) term and show that asymptotic expansion converges for all $\epsilon$ in a neighborhood of zero. Does this asymptotic expansion converge to $x(\epsilon)$ for $\epsilon\neq 0$?
Some1 please help :(