Integrating Power Series for a Function

[quasar_4]

Homework Statement

Find a power series representation for the given function using termwise integration.

$$f(x) = \int_{0}^{x} \frac{1-e^{-t^2}}{t^2} dt$$

Homework Equations

The Attempt at a Solution

Well, I figured I could rewrite it like this using the Maclaurin series for exp(-x) (plugging in t^2 for x):

$$\int_{0}^{x} \frac{1}{t^2} - \frac{\sum_{n=0}^{\infty} \frac{(-1)^n (t^2)^{2n}}{n!}}{t^2} dt$$. The series term in the integral works out fine, but the problem is that then my integral has the term 1/t^2, which integrates to -1/t, and I'm supposed to evaluate this from 0 to x, which is clearly bad at 0. What am I doing wrong? Is it the Maclaurin series for exp(-x)?

 

[Mark44]

The first term in your Maclaurin series for e^-t2 is 1, right? So 1 - e^-t2 is just going to be all the other terms of the series, but with opposite signs.

 

[quasar_4]

Ha! That makes me feel very silly.

I was just thinking too hard...

Thank you!