A Can You Prove the Series Is Periodic?

[vela]

Anyway, now I'm confused. Can you explain, regardless of how many values T can take (because it's value wasn't specified), why it isn't iff in either interpretation? Is it the "if" part you don't like, or the "only if" part?

If you define T as the smallest value, then if T is "the" period, we know f(x)=f(x+T) for all x, but if some value T satisfies f(x)=f(x+T) for all x, that doesn't imply T is "the" period but rather just "a" period. So it comes down to how one defines T. I've always seen it defined as the smallest value that satisfies f(x)=f(x+T) for all x, but apparently, not everyone defines it that way.

 

[PeroK]

If you define T as the smallest value, then if T is "the" period, we know f(x)=f(x+T) for all x, but if some value T satisfies f(x)=f(x+T) for all x, that doesn't imply T is "the" period but rather just "a" period. So it comes down to how one defines T. I've always seen it defined as the smallest value that satisfies f(x)=f(x+T) for all x, but apparently, not everyone defines it that way.

In this case, can you prove that $2\pi$ is the fundamental period of the function in question? It's easy to see that it does not have period $\pi$. But, it might be tricky to show that it can't be anything else less than $2\pi$.

 

[vela]

In this case, can you prove that $2\pi$ is the fundamental period of the function in question? It's easy to see that it does not have period $\pi$. But, it might be tricky to show that it can't be anything else less than $2\pi$.

I'd offer the physicist's proof: Look at the graph. ;)

 

[Office_Shredder]

In this case, can you prove that $2\pi$ is the fundamental period of the function in question? It's easy to see that it does not have period $\pi$. But, it might be tricky to show that it can't be anything else less than $2\pi$.

I believe it's a theorem that either the fundamental period is $2\pi/n$ for some $n$, or the function has arbitrarily small periods. The second one can't be true because the function is continuous and not constant (probably? Maybe this is a trick question haha). The first one doesn't feel impossible to manage but I agree isn't easy.

Edit to add: it is not constant, since $f(0)=0$ and $f(\pi/2)=1$