For a complex power series $$P(z)=a_0+a_1z+a_2z^2+...,$$ $P(z)$ converges if $$\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}z|=|z|\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|<1.$$ The radius of convergence $R$ is defined by $${1\over R}=\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|.$$ If we show this radius of convergence on an Argand diagram by drawing a circle of radius R with centre at the origin, $P(z)$ is convergent for $z$ values lying inside the circle.For $z$ values lying outside the circle, $P(z)$ diverges. For $z$ values lying on the circle, the series may converge or diverge.
My question is why can the series converge for $z$ values lying on the circle? I know that for a geometric series in the form of $a+ar+ar^2+...$ the series can only converge for $|r|<1$. For z values lying on the circle, it is analogous to saying that $|r|=1$ so the series should not converge.
Edit: I came across this on a textbook and had posted a screenshot of it here.
