Find the power series representation?

Question

$$f(x)=xe^{x^2}$$

$$f(x)= \sum_{n = 0}^{\infty} \frac1{an!}x^{pn}$$

Need to find $an$ and $pn$?

Not sure how to approach this one, even though I have been solving these type of questions before.

Answer 1

You know the Taylor Series for $e^x$:

$$e^x = \sum_{n=0}^\infty {1\over n!} x^n$$

And when multiplying by $x$ and plugging in $x^2$, we get:

$$xe^{x^2} = \sum_{n=0}^\infty x\cdot {1\over n!} \left(x^2\right)^n = \sum_{n=0}^\infty {1 \over \frac 1x n!} x^{2n}$$