So, by Taylor's theorem, given a holomorphic function at a point, we can calculate the Taylor series for f of following way: $$f(z) = f(a) + f'(a) (z-a) + \frac{f''(z)}{2!} (z-a)^{2} + ... + \frac{f^{(n)}(z)}{n!} (z-a)^{n} + .... $$

This leads us to a more general type of series of functions called power series, wich is the study of series of the form: $$\sum _{n=0}^{\infty}a_{n}(z-a)^n$$ Note at first that this kind of series are always convergent for z=a.

The previous series converges in some points and diverges in others, in general, there exists a radius R such that the series converges if |z-a| < R and diverges if |z-a| > R. For those points such that |z-a| = R, the series may or not converge. Extreme cases are considered in which R = 0 when the series only converges for z = a or R = ∞. In general, R is called the radius of convergence of the power series.

We show below some tests of convergence for power series:

### Comparison test

If series \(\sum |v_{n}| \) is convergent and \( |u_{n}| \le |v_{n}| \), then the series given by \(\sum u_{n} \) is absolutely convergent.On the other hand, if the series \(\sum |v_{n}| \) diverges and \( |u_{n}| \ge |v_{n}| \), then the series \(\sum |u_{n}| \) diverges, but series \(\sum u_{n} \) can converge or not.

### Quotient test

If \( \lim_{n\to \infty} |\frac{u_{n+1}}{u_{n}}| = L\), then \(\sum u_{n} \) converges (absolutly) if L < 1 and diverges if L> 1. If L=1 the test fails.### Root test

If \( \lim_{n\to\infty} \sqrt[n]{|u_{n}|} = L \), then \(\sum u_{n} \) converges (absolutly) if L < 1 and diverges if L> 1. If L=1 the test fails.There are some other tests, integral, of Gauss, but with these for now it will be enough for the examples that we will develop here. The following is a test for uniform convergence: