I am trying to prove that $f(x) = x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + ...$ is continuous for a fixed $x\_0 \in (-1,1)$ by using the Weierstrass M-test. Now, the solution in the book is not what I expected, and not even close to what I had in my own mind. So, can some one verify if I am totally of with my approach to solve this

Here was my suggestion:

Let $M\_n = x^n$, then for each $f\_n(x) = \frac{x^n}{n}$ we have that $f\_n(x) \leq M\_n$. We know that $\sum\_{n=1}^{\infty} M\_n$ converges since it is a geometric series, hence $\sum\_{n=1}^{\infty} f\_n(x)$ converges uniformly by Weierstrass M-test, and by the Continuous Limit Theorem f(x) is continuous at $x\_0$.

I guess there is a mistake here somewhere...could anybody point it out for me?