Problem 1 is the problem I received the least amount of points on, therefore I will solve it.
Is the signal
$ x(t) = \sum^{\infty}_{k = - \infty} \frac{1}{(t+2k)^2 + 1)} $
periodic? Answer yes/no and justify your answer mathematically.
Yes, because:
$ x(t+2) = \sum^{\infty}_{k = - \infty} \frac{1}{(t+2+2k)^2 + 1} $
$ = \sum^{\infty}_{k = - \infty} \frac{1}{(t+2(k+1))^2 + 1} $
let r = k + 1
$ = \sum^{\infty}_{r = - \infty} \frac{1}{(t+2r)^2 + 1} = x(t) $
