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 $b$
Since the function has Taylor series that converge to that function at every point, as Mittens pointed out (and as a consequence of Abel's theorem), we have that the series of functions, which Taylor seires is, converges uniformly to that function at every point. Then, by Integrable Limit theorem and Integral of sum theorem:
$$\int_a^b \lim\limits_{N \to \infty}\sum_{n = 1}^N f_n = \lim\limits_{N \to \infty}\int_a^b \sum_{n = 1}^N f_n = \lim\limits_{N \to \infty} \sum_{n = 1}^N \int_a^b f_n$$