The question
Is the signal $ x(t) = \sum_{k = -\infty}^\infty \frac{1}{(t+2k)^{2}+1} $ periodic?
$ x(t+2) = \sum_{k = -\infty}^\infty \frac{1}{(t+2+2k)^2+1}\, $
$ x(t+2) = \sum_{k = -\infty}^\infty \frac{1}{(t+2(k+1))^2+1}\, $
Subsitute $ r $ = $ k+1 $
$ x(t+2) = \sum_{r = -\infty}^\infty \frac{1}{(t+2r)^2+1} = x(t)\, $
Clearly, the system is periodic.
