Infinite sum of analytic functions becomes non-analytic

Question

I am looking for a simple example that a series of analytic functions can become non-analytic. This is in the context of phase transitions, where one considers the analyticity of the partition function which may show singular behaviour in infinitely large systems, signaling a phase transition …

Many thanks!

Answer 1

Consider $f_n(z)=z^n$.

$f_n(z)$ is an entire function for every integer $n\ge0$.

However, $$\sum^\infty_{k=0}f_k(z)$$ only converges for $|z|<1$. In other words, this sum of entire functions is not analytic outside the unit disk.