I'm studying physics and am currently following a course on complex analysis and in the section on analytic functions, the radius of convergence $R$ for power series was introduced. The Taylor expansion around $z_0=0$ for the exponential function was considered as an example of a power series with $R\rightarrow\infty$. The notes state this can be proved by using Weierstrass' Criterion for uniform convergence, which I'll state in my own words:
Consider a series
$\sum\limits_{k=0}^{\infty} f_k(z)$.
If you know numbers $a_k$ for which
$|f_k(z)| < a_k$
for all $z$, and
$\sum\limits_{k=0}^{\infty} a_k$
converges uniformly, then also
$\sum\limits_{k=0}^{\infty} f_k(z)$
converges uniformly.
For the exponential, we have the power series
$e^z = \sum\limits_{k=0}^{\infty}\dfrac{z^k}{k!}$.
Now I've been thinking about this, but I can't seem to think of a uniformly converging series of $a_k$'s that bound the terms of this power series. Perhaps this is really straightforward and I wouldn't have any difficulties with it if I remembered my course on real analysis a bit better…
It's not a homework problem and series convergence is not a main goal in this course, but it's been bugging me that I don't understand why Weierstrass's Criterion proves that the radius of convergence goes to infinity for the exponential, so I thought I'd ask here. Thanks in advance.
Best Answer
You need to be careful in making the distinction between uniform convergence for any $z \in \mathbb{C}$ and uniform convergence for $|z| < R$ for a fixed $R$.
The exponential function is unbounded. If we were able to find a sequence $a_k$ so that for any $z \in \mathbb{C}$ $$ \left|e^z\right| = \left|\sum_{k=0}^\infty \dfrac{z^k}{k!}\right| \le \sum_{k=0}^\infty \left|\dfrac{z^k}{k!}\right| \le \sum_{k=0}^\infty a_k < \infty $$
then the exponential function would be bounded, a contradiction.
Now, if we limit the domain of the exponential function to $|z| < R$ for a fixed $R > 0$, then: $$ \sum_{k=0}^\infty \left|\dfrac{z^k}{k!}\right| \le \sum_{k=0}^\infty \dfrac{R^k}{k!} < \infty $$
The convergence of $\displaystyle \sum_{k=0}^\infty \frac{R^k}{k!}$ can be established via the ratio test. Thus, the exponential function is uniformly convergent for any fixed $R > 0$ no matter how big it is.