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Periodicity of Shifted Sum

I struggle to understand how to prove whether or not a signal is periodic if the signal is written as the sum of shifted functions. A good example of this type of problem would be problem number 1 from the recent exam:

$ x(t) = \sum_{k=-\infty}^{\infty}\frac{1}{(t+2k)^2 + 1} $

Is the signal periodic?

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