I want to justify moving the limit inside of this integral:
$$\lim_{n \to \infty}\int_{0}^{n}\sum_{k=0}^{n}(-1)^k\frac{x^{2k}}{(2k)!}e^{-2x}$$
My idea is to use DCT. The function inside is dominated by $e^{-x}$. By taking absolute values we get the inside is always smaller (on $[0,\infty)$) than $$\sum_{k=0}^{\infty}\frac{x^{2k}}{(2k)!}e^{-2x}\leq e^{-x}$$ Where the inequality comes from the power series of $e^x$. Thus we can apply DCT.
Is this correct?