why we don't have : $\lim_{z \to 1} f(z) = \sum_{n= 0}^\infty a_n$

Question

Let's consider a power series : $\sum a_nz^n$ with radius of convergence $R \geq 1$.

Moreover let's denote :

$$C = \{z \in \mathbb{C} \mid \mid z \mid \leq 1 \}$$ Now let's define :

$$f(z) = \sum a_nz^n, \forall z \in C$$

Now, we know that $f$ is $\mathbb{C}^{\infty}$, so it's continuous on $C$.

So in this case why don't we have :

$$\lim_{z \to 1, z \in C} f(z) = \sum_{n = 0}^\infty a_n $$

Normally by the continuity of $f$ this is always true ?

Answer 1

Note that saying the radius of convergence is $ R =1$ means the power serie converges for all $\mid z \mid < 1$.

So here using the continuity doesn’t mean anything because we can’t say anything for $z =1$.

Your equality is true if : $R > 1$. You can also use Abel theorem to proove that your equality is true when you have some extra conditions.

Answer 2

A power series may not converge on the border of its disk f convergence.

Simple example

The power series $$f_1(z) = \frac{1}{1-z} = \sum\limits_{n \ge 0} z^n$$ is convergent on the unit disk $\mathbb U$ and therefore continuous at every point of that disk. However the power series diverges at $z=1$.

More funny example

I give an example of a A POWER SERIES CONVERGING EVERYWHERE ON ITS CIRCLE OF CONVERGENCE DEFINING A NON-CONTINUOUS FUNCTION in my math counterexamples blog.