I have a trancendental eq.

x/(x-1) + exp{(x-1)/ε} - ε = 0

I need to derive an asymptotic expansion for any bounded roots x(ε) this equation might have. Find the coefficient of the nth (‘general’) term and show that asymptotic expansion converges for all ε in a neighborhood of zero. Does this asymptotic expansion converge to x(ε) for ε=/0?

And does the equation above also have unbounded roots? If it does, can we derive asymptotic expansions for them?