If a power series converges on some interval centered at the center of convergence, then the distance from the center of convergence to either endpoint of that interval is known as the radius of convergence which we more precisely define below.
I am a bit confused as to whether I am doing this question correctly.
Firstly, we have defined the radius of convergence of a power series centered at a $$sum_{n=0}^{infty} a_n(x-a)^n$$
to be the positive real number $R$ such that the power series converges uniformly on the interval $(a-R,a+R)$ and for $x lt a - R$, $x gt a + R$, the series does not converge.
So, we must show that the radius of convergence of the Taylor series of $e^x$ centered at 0 is infinite. Initially, I answered showing it converged using the ratio test, however, I believe this is wrong as it would only show that the series converges, not necessarily uniformly, correct?
I was wondering if it would be correct to show that for the series $f_n(x) = frac{x^n}{n!}$ that the series $sum_{n=0}^{infty} frac{x^n}{n!}$ converges absolutely and uniformly on the bounded interval $(-a,a)$ for any $a in Bbb{R}$ using the Weierstrasse M-test (by taking $M_n = frac{a^n}{n!}$ and showing that the sum of this series converges by the ratio test) and then conclude that because I can choose any $a in Bbb{R}_0^+$ for which the series converges uniformly, the radius of convergence is infinite (ie. I can choose an interval of any size for the M-test), or do I have to show that the series converges uniformly specifically on the interval $(-infty,infty)$?
Thanks for your help!
2 Answers
$begingroup$Hint: $R = displaystyle lim_{n to infty} dfrac{|a_n|}{|a_{n+1}|}$, and $a_n = dfrac{1}{n!}$.
DeepSeaDeepSea![Radius Radius](https://i.ytimg.com/vi/r8bK9j4vKco/maxresdefault.jpg)
You can use Cauchy-Hadamard theorem to calculate the radius of convergence.$$ frac{1}{r} = limsup_{j to infty} |a_j|^{1/j}$$For Cauchy sequences, $limsup$ can be replaced by $lim$. Then, since the two following limits agree$$ lim_{j to infty} |a_j|^{1/j} = lim_{j to infty} frac{|a_{j+1}|}{a_j}$$Using the ratio test, it can be shown that $r^{-1} = 0$.
![Radius Radius](https://d2vlcm61l7u1fs.cloudfront.net/media%2F6b6%2F6b687115-dd60-4fd8-9db9-90fe37892a47%2Fphpe1EBQr.png)