My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence.
I am comfortable with the fact that if the limit of the function at a certain point $a$ doesn't exist, the power series won't converge for this value of $x$. But why can't the series converge for $x$ whose absolute value is greater than this point? Why aren't there "areas of convergence" instead of a single radius of convergence?
Best Answer
Assume the series $\sum_{k=0}^\infty a_kz^k$ converges for some $z\in{\mathbb C}$. Then it converges absolutely for any $z'$ that is nearer to the origin than $z$; see the proof below.
From this fact it follows by mere logic, that when the series diverges at some $z\in{\mathbb C}$ it cannot converge at any point $z''$ farther away from the origin.
Proof. When $\sum_{k=0}^\infty a_kz^k$ converges at $z$ then there is an $M$ with $\left|a_kz^k\right|\leq M$ for all $k$. Assume now that $0\leq r<\left|z\right|$. Then $q:={r\over\left|z\right|}<1$ and $$\left|a_k\right|r^k=q^k\left|a_k z^k\right|\leq Mq^k\qquad(k\geq0)\ .$$ It follows that the series $\sum_{k=0}^\infty\left|a_k\right|r^k$ converges.