# The radius of convergence for a power series and the existence of the limit

I found a definition for the radius of convergence of a power series: $$r =\ \lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}}\right|$$

or $$r =\ \lim_{n \to \infty}\frac{1}{\sqrt[n]{|a_n|}}$$ if the limit exists .

I know that this comes from the ratio and root test , and I solved many problems about radius of convergence..

But I would like to know what is the meaning of " the limit exists " here.

Does it mean that if the limit is infinity then there is NO radius of convergence , because it will converge for all values of x ? (However , my teacher said that if limit is infinity then the radius of convergence exists and is equal to infinity .

OR it means that in some cases , we will find the limit has more than one value or we can not find its value ( like if we take the limit for sin(n) )

• A power series has always a radius of convergence $R\in [0,+\infty]$. – hamam_Abdallah Dec 19 '17 at 19:27
• So the " limit does not exist " here just means that we can not find a specific value for the limit ( like if we take the limit for sin(n) ) ? – MCS Dec 19 '17 at 19:33
• Even if the limit does not exist, the radius exists. – hamam_Abdallah Dec 19 '17 at 19:37
• The purpose of these tests is to make a comparison with a geometric series. math.stackexchange.com/questions/2202516/…. – zwim Dec 19 '17 at 19:38
• According to Terence Tao, think in limsup and liminf, which always exists. (can be finite or infinity) "Limit exists" can thus be translated to "liminf=limsup". – GNUSupporter 8964民主女神 地下教會 Dec 19 '17 at 19:52

"The limit $\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}}\right|$ exists" in this context means
• the sequence of real number $\{| \frac{a_{n}}{a_{n+1}}|\}$ is convergent: there exists a real number $L$ such that $$\forall\epsilon>0\ \exists N\in{\bf N}\ \forall n\geq N\ \ |\frac{a_{n}}{a_{n+1}}-L|<\epsilon$$
• or $| \frac{a_{n}}{a_{n+1}}|\to\infty$ as $n\to\infty$.
(The other one this similar.) In the second case, it is a convention to say that the raidus of convergence is $\infty$, although $\infty$ is not a real number.
Given a sequence of complex numbers $\{a_n\}$, the two limits $$\lim_{n \to \infty} \left| \frac{a_{n}}{a_{n+1}}\right|, \quad \lim_{n \to \infty}\frac{1}{\sqrt[n]{|a_n|}}$$ may or may not exist. If one of them (or both) exists, then it is defined as the the radius of convergence of the corresponding power series.