My question is related to this one. Suppose we are trying to solve the following integral $$ \int f(x) g(x) dx,$$ where we know $f(x)$ is smooth, all of its derivatives are positive, and the sum of its derivatives $\sum_{n=0}^{\infty} f^{(n)}(x)$ converges. For the sake of simplicity, let's say $g(x)=e^x$ for now. Then if we naively apply integration by parts "infinitely times" with $u=f(x)$ and $v'=g(x)=e^x$ we get $$ \sum_{n=0}^{\infty} (-1)^n e^x f^{(n)}(x) = e^x \sum_{n=0}^{\infty} (-1)^n f^{(n)}(x). $$ Since the above series converges (by absolute value test and our assumption), were we justified in performing integration by parts infinitely times? In other words, is the condition that a series of a function's derivatives (which are all positive) converges, sufficient to performing integration by parts infinitely times (with $g(x)=e^x$)?
Now what about for a more general function $g(x)$? What conditions are needed on $g(x)$ to allow infinite applications of integration by parts (with the same assumptions on $f(x)$)? Thanks to all in advance.