The questions is, find: $$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]$$ I started by saying: $$\mathcal{L}\left[\sum_{n=0}^\infty(-1)^n\text{H}(t-n)\right]=\int_0^\infty\sum_{n=0}^\infty(-1)^n\text{H}(t-n)e^{-st}dt=\sum_{n=0}^\infty(-1)^n\int_0^\infty\text{H}(t-n)e^{-st}dt$$ $$=\sum_{n=0}^\infty\frac{(-1)^ne^{-ns}}{s}$$ My questions are:
- Can this summation be simplified in any way, or is this the best form?
- In this context I assume it is same to interchange the integral and summation, but are there any contexts where an integral and summation are NOT interchangeable?
Thanks
EDIT: I found this post talking about changing integrals and summations, but I am not sure I fully understand
