Let us say there is a function $f(x)$. Let us say that that it has a Fourier (co)sine series representation $$g(x) = \sum_{n=1}^{\infty} a_n\sin(kx) = f(x)$$
I am having difficulty understanding a question that asks me to find the value to which $g(x)$ converges to over an interval $[a, b]$? $g(x)$ is a function, so it cannot converge to a single value over an interval?
Recall that $\sum_{n=1}^\infty a_n \sin(kx) = \lim_{N\to \infty} \sum_{n=1}^Na_n \sin(kx)$ is indeed a limit, and we say the sum converges if the limit exists. To me, it looks like the question is asking the following:
Suppose $f$'s fourier sine coefficients are $(a_n)_n$. For what values of $x$ does the series converge $\sum_{n=1}^\infty a_n \sin(kx)$ converge? Call this set $D$. Then for what $x \in D$ do we have $f(x) = \sum_{n=1}^\infty a_n \sin(kx)$?
The subtlety here is that in general, the fourier series of a function does not necessarily converge to that function. The question is asking you to figure out when the series actually converges to what you would hope it converges to.