Question 1 was the hardest one for me. I knew there was a trick, but I just couldn't remember how to do it.
Question 1
Is the signal
$ x(t)=\sum_{k=-\infty}^{\infty} \frac{1}{(t+2k)^2+1} $
periodic? Answer yes/no and justify your answer mathematically.
Answer
Yes. Because if you solve for x(t+2) then:
$ x(t+2)=\sum_{k=-\infty}^{\infty} \frac{1}{(t+2+2k)^2+1} =\sum_{k=-\infty}^{\infty} \frac{1}{(t+2(k+1))^2+1} $
Let r=k+1 then:
$ x(t+2)=\sum_{k=-\infty}^{\infty} \frac{1}{(t+2r)^2+1}=x(t) $
Therefore because x(t+2)=x(t) the signal is periodic.
