How Can I Find the Power Series Representation of the Given Integral Function?

[girolamo]

Hi, I'm trying to find the series representation of $$f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt$$. I have found it ussing the Maclaurin series, differenciating multiple times and finding a pattern. But I think it must be an eassier way, using the power series of elementary functions. I know that $$e^{x}=\sum_{0}^{\infty}\frac{x^{n}}{n!}$$ and $$\frac{1}{1+x}=\sum_{}^{\infty}(-1)^{n}x^{n}$$ but I don't know how to use it here. Thanks

(Don't hesitate to correct my english)

 

[pwsnafu]

If $f(x) = \sum_{n=0}^\infty a_n x^n$ then
$\frac{f}{1-x} = \sum_{n=0}^\infty \sum_{j=0}^n a_j x^n$.