I've found some examples when $\sum\int \neq \int\sum$ but I'm interested in how 'likely' I am to get the wrong answer if I do this manipulation when solving integrals, which often have you integrating the Taylor series of a function termwise.
Is there a real analytic function $f(x)$ with its Taylor expansion $f(x) = \sum_{k=0}^\infty{T_k(x)}$ such that $\int_{a}^{b} f(x) \neq \sum_{k=0}^\infty{\int_a^b{T_k(x)}}$ for some $a$ and $b$