I was solving an exercise and was stuck on the following problem:
I am given a periodic, continuous function $f(t)$ and I know that it has a fourier series representation of $$f(t) = \sum_{n=-\infty}^\infty a_n e^{j\frac{2\pi}{T}nt}$$ for a sequence $\{a_n\}$, where $T$ is the period of $f(t)$. I am asked to find the fourier series coefficients of a function $g(t)$, where $$g(t) = \int_{-\infty}^t f(x)dx$$
My first question arises here: How can I know if the expression provided for $g(t)$ converges? Do I have to accept something for simplicity?
Not caring about this and reading more, I see that the problem also states that I can assume that the average value of $f(t)$ is $0$. By this, I think they meant something that simplifies everything nicely, but I understand the following: $$\int_a^{a+T}f(x)dx = 0, \forall a\in\mathbb{R}$$ I still did not think that this assumption solved any issues, but it helps me see that the period of $g(t)$ must also be $T$.
I went on with writing the analysis formula. But everything I obtained seems senseless to me, so as the issue I had at the beginning of the question. What am I missing?
P.S. I am new to this topic and area, so please provide extra information for your notation or if you see that I don't know something. Thank you!