Fourier series of a function

The exercise states "Let be a periodic function with period defined in the interval by if and if .

Calculate the Fourier coefficients of f and obtain its Fourier series and analyze the convergence of the series.

My attempt: First of all I think they made an error and meant to say the interval . Otherwise it's not making sense to me.

Ok so I read the definition of the nth coefficient of the Fourier series in my class notes and I read .

So I calculated to be worth .

For all even and , I found out that . While for all uneven, I found out that .

Since the Fourier series is defined as , I reach that the Fourier series is worth where with , .

To check out the convergence of the series, I guess I should split the series in 2, starting from to and from to , and then apply the quotient test?

Is what I've done right? I'm not confident but can't find any error.

Edit: I forgot to multiply by "i" for the series of the uneven terms I think. I don't think the result change that much, except with an "i" factor.