Asymptotic Expansion: First 2 Terms for Integral

[the_kid]

Homework Statement

Find the first two terms in an asymptotic expansion of the following as x goes to zero from the right (i.e. takes on smaller and smaller positive values).

\int^{1}_{0}e^{-x/t}dt

Homework Equations

The Attempt at a Solution

I'm not exactly sure how to proceed with this. I'm assuming I should expand the integrand as a Taylor series:

e^{-x/t}=\sum^{\infty}_{k=0}\frac{(-x/t)^{k}}{k!}

I'm kinda stuck here, though.

 

[STEMucator]

Notice you start having problems as t goes to zero... you should re-write your integral with a limit and then I think it would be best to actually expand the terms of your sum this time.

 

[the_kid]

Notice you start having problems as t goes to zero... you should re-write your integral with a limit and then I think it would be best to actually expand the terms of your sum this time.

lim_{a\rightarrow0^{+}}\int^{a}_{0}(1-\frac{x}{t}+\frac{x^{2}}{2t^{2}}-\frac{x^{3}}{6t^{3}}+\cdots)dt

Like this? Hmm...