The recent post here has led to the following question (consider $\cos(sx) = \sum\nolimits_{k = 1}^\infty {\frac{{( - 1)^k (sx)^{2k} }}{{(2k)!}}}$). I find it instructive, whether it will turn out to be easy/well-known or not.
So, suppose that $(f_k)_{k \geq 1}$ is a sequence of continuous functions on a non-compact interval $I$. Suppose that $ \sum\nolimits_{k = 1}^\infty {f_k (x)}$ converges uniformly on every compact subinterval of $I$. Further, suppose that both $\sum\nolimits_{k = 1}^\infty {\int_I {f_k (x)\,{\rm d}x} }$ and $\int_I {[\sum\nolimits_{k = 1}^\infty {f_k (x)]} \,{\rm d}x}$ converge (i.e., are finite).
Under these assumptions, can you find an example where $$ \int_I {\bigg[\sum\limits_{k = 1}^\infty {f_k (x)\bigg]} {\rm d}x} \neq \sum\limits_{k = 1}^\infty {\int_I {f_k (x)\,{\rm d}x} }? $$ Consider also the case where we are only given that the right-hand side converges.
If relevant: What about the case where $\sum\nolimits_{k = 1}^\infty {f_k (x)}$ and its partial sums are analytic functions?
Any relevant discussion is welcome here.