# Integrating power series

1. Oct 7, 2009

### quasar_4

1. The problem statement, all variables and given/known data

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

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

2. Relevant equations

3. 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)?

2. Oct 7, 2009

### Staff: Mentor

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.

3. Oct 7, 2009

### quasar_4

Ha! That makes me feel very silly.

I was just thinking too hard... :rofl:

Thank you!