Improper Integral with product of series

Question

I am trying to solve $$ \int_0^\infty \left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6 }+\cdots\right)\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2 }+\cdots\right)\text dx$$ but failed, even the left hand side can be expressed as $x\cdot e^{-x^2 /2}$, I can still not deal with the second parenthesis. Is there anyway to evaluate this integral when integrand involved series which is a sum of squares ?

Answer 1

Rather than knowing the series on the right outside of a general term, try distributing what you do know and make another series that you can integrate with.

I know it seems vague, but that's how I got the answer and I figured you want to solve this on your own.

If you want me to explain what I did, I can.