Question (1) What are the conditions the complex function $f\_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with finite integration is possible? 


$$\int\_1^B\sum\_{1}^{N} f\_n(t)dt = \sum\_{1}^{N} \int\_1^B f\_n(t)dt .$$


Question (2) After we take the limits of on both sides of the equation above, do we get the same limits?


$$\lim\_{B\rightarrow\infty}\lim\_{N\rightarrow\infty}\int\_1^B\sum\_{1}^{N} f\_n(t)dt = \lim\_{N\rightarrow\infty}\lim\_{B\rightarrow\infty}\sum\_{1}^{N} \int\_1^B f\_n(t)dt .$$


Thanks-
Mike



a question regarding the interchange the order of finite summation with finite integration
1) See Fubini's Theorem. You want $\sum\_1^N \int\_1^B |f\_n|$ to be finite. 
This is just the case where one of the two measures is counting measure.


2) Again, Fubini's theorem, this time with $\sum\_{1}^\infty \int\_1^\infty |f\_n|$.


