Interchanging summation and integration I recently came across a question which i posted here Integration twist
Now i have no idea when to interchange summation and integration which involves a different concept (as proposed by a person in comments). Can anyone please discuss the concept involved ?
 A: You can interchange integration and summation if the series converges uniformly.  There are other, weaker conditions, but this will do for the present case.  In the the question, you were integrating a power series.  Since a power series converges uniformly on any compact interval within the interval of convergence, on such an interval the power series can be integrated term-by-term.
In the example, the interval on integration is $[0,1]$ and the interval of convergence is $(0,1)$.  The series doesn't even converge at $1$.  All is not lost, however.  The series converges uniformly on $[0,a]$ of any $0<a<1$.
Now we can say $$\begin{align}
\int_0^1\sum a_nx^n\,\mathrm{d}x&=\lim_{a\to1-}\int_0^a\sum a_nx^n\,\mathrm{d}x\tag1\\
&=\lim_{a\to1-}\sum \int_0^aa_nx^n\,\mathrm{d}x\tag2\\
&=\lim_{a\to1-}\sum a_n\frac{a^{n+1}}{n+1}\tag3\\
&=\sum \frac{a_n}{n+1},\tag4
\end{align}$$
provided the sum converges.

*

*First fundamental theorem of calculus.

*Uniform oncvergence on $[0,a]$

*Perform the integration

*Abel's limit theorem.

Fubini's theorem is a more advanced theorem, which I would advise you not to try to use until you have learned something about measure theory.
