How to show that the series of functions $\sum_{k = 0}^\infty(-1)^k \frac{x^{2k + 1}}{2k + 1}$ is pointwise convergent to a function?

Question

The problem I'm given is as follows:

Consider the sequence of functions $(f_k)_k$, where $f_k(x) = (-1)^k \frac{x^{2k + 1}}{2k + 1}$ as well as the resulting series: $$A(x) = \sum_{k = 0}^\infty f_k (x)$$ Show that the series converges pointwise towards a function $f(x)$ for all $x \in (-1, 1)$. It is not yet necessary to show which $f$ the series converges to.

My confusion in answering this question arises from the fact that the tests (Cauchy and M-Test) determine uniform convergence, which is stronger than pointwise. I understand that passing either test would be necessary and sufficient for uniform convergence and therefore also for pointwise, but since the question asks only to show pointwise convergence, I'm a bit lost.

My approach is as follows:

1. By the ratio test, I have shown that the series converges absolutely (to a value) for all $x \in (-1, 1)$.

2. The next step should be to somehow justify that the values for each $x$ that the series converges to belong to a function, but I am not sure how to do that.

If anyone could help me to structure the last step, I'd be very grateful.

Answer 1

If we really have to constraint ourselves to the minimal level of math knowledge needed. Since you mentioned no integral, I assume no Taylor series either, so there is no way for you to tell what $f(x)$ is, but you do not seem to need that.

The most direct way is the ratio test, which proves the absolute convergence of the series. May be this suffice?