Hey there!

I have a doubt in the notation of a problem involving series that I am trying to solve.

Are integrals and summations exchangeable? I have something like this:

$\displaystyle \int_0^t\sum\limits_{k=1}^\infty B_k\left(s\right)\sin\left(\frac{{k\pi}}{{L}} x\right)e^{-\alpha\left(k\pi/L\right)^2\left(t-s\right)}ds$

Is it the same as

$\displaystyle \sum\limits_{k=1}^\infty \int_0^t B_k\left(s\right)\sin\left(\frac{{k\pi}}{{L}} x\right)e^{-\alpha\left(k\pi/L\right)^2\left(t-s\right)}ds$

I think that for solving this is much more convenient to integrate the general term first in terms of k, and then evaluate the sum.

Any hint?