Term by term integration of series on infinite intervals

Question

Suppose $f(x)=\sum a_n x^n$ is given by its Taylor series such that $\int_0^{\infty} f(x)dx$ exists.

Under what conditions can we integrate term by term by replacing $f(x)$ by its Taylor series?

Answer 1

Inside the radius of convergence, a power series may be integrated term-by-term. This should be an all calculus textbooks.

I am assuming that you want to get an indefinite integral for $f$.

Example: $e^x = \sum_{n=1}^\infty\frac{1}{n!} x^n$, integrate term-by term to get $\sum_{n=1}^\infty \frac{1}{n!}\;\frac{1}{n+1} x^{n+1} = e^x-1 = \int_0^x e^t\;dt$.

(As @zhw. noted, you never get $\int_0^\infty$ this way, except for all $a_n = 0$.)