Okay, so I'm looking at the anwers to a question where you're supposed to solve a definite integral depending on $x$. And I do not understand the equality below:
$$\int_{0}^{1} \frac{t^2}{1-tx} dt = \sum_{n=0}^{\infty} (\int_{0}^{1} t^{2+n} dt)x^n$$
I know about the sum of the power series, but I still can't understand the equality.
Does anyone care to explain?
Thanks
Just use the fact that$$\frac{t^2}{1-tx}=t^2\sum_{n=0}^\infty(tx)^n=\sum_{n=0}^\infty t^{n+2}x^n,$$if $\lvert x\rvert<1$ (and $t\in[0,1]$).