Finding a power series expansion for a definite integral

[Szichedelic]

Homework Statement

Find a power series expansion about x = 0 for the function

f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds

Homework Equations

The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k}

The Attempt at a Solution

I've tried several things to start off with but quickly end up hitting a dead end road. First, I tried just simply taking the integral, but quickly found it isn't defined at 0 (hence why they are asking me to find a power series expansion for it). Then, I tried finding a power series expansion for the innerpart of the integral and ran into the same problem.

 

[LCKurtz]

Homework Statement

Find a power series expansion about x = 0 for the function

f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds

Homework Equations

The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k}

The Attempt at a Solution

I've tried several things to start off with but quickly end up hitting a dead end road. First, I tried just simply taking the integral, but quickly found it isn't defined at 0 (hence why they are asking me to find a power series expansion for it). Then, I tried finding a power series expansion for the innerpart of the integral and ran into the same problem.

Did you try substituting the Taylor series (as a function of x) for $e^{-sx}$ in the integrand, simplifying and integrating?

 

[Szichedelic]

Yeah, I figured that out shortly after I posted this. Can't believe I overlooked that! Thanks!