Is this function/series periodic?

Question

$$f(t)=\sum_{k=-\infty}^{\infty}(-1)^kp_{0.5}(t-2k)$$

Recall: $$p_{\Delta}=\begin{cases}\frac{1}{\Delta},&0\leq t\leq\Delta\\0&\text{ otherwise.}\end{cases}$$

Is the function periodic? If so, what's the fundamental period?

Intuitively, I don't think it is. I think each term would cancel out the previous term because of them being opposite signs. I have no clue how to prove it though. Can someone please give me a hand?

Answer 1

Show that

• this expression ideed defines a function because for any specific $t$, there are only finitely many $k$ for which the $k$th summand is nonzero.
• $f$ is continuous except when $t\in 2\mathbb Z$ or $t\in \frac12+2\mathbb Z$; conclude from this that any period of $f$ must be a multiple of $2$.
• $f(t+2)=-f(t)$ (by reindexing the summands); hence $f$ is identically zero or has fundamental period $4$