A community for students.
Here's the question you clicked on:
 0 viewing
Loser66
 one year ago
Find radius of convergence of
\(\sum_{n=0}^\infty \dfrac{5^n}{n!} z^n\) . I got infinitive am I right?
Please, help
Loser66
 one year ago
Find radius of convergence of \(\sum_{n=0}^\infty \dfrac{5^n}{n!} z^n\) . I got infinitive am I right? Please, help

This Question is Closed

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1Is this the formula for the radius of convergence? \[ \limsup_{n\to\infty}\sqrt[n]{\frac{5^n}{n!}}z \]

freckles
 one year ago
Best ResponseYou've already chosen the best response.0what about ratio test

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1I thought he wanted to show that the radius of convergence is infinite. I have not taken any analysis course but Wikipedia says that ratio test only shows that the radius of convergence is finite.

freckles
 one year ago
Best ResponseYou've already chosen the best response.0\[n \rightarrow \infty \\ \frac{a_{n+1}}{a_n}<1 \\ \frac{5^{n+1}}{(n+1)!}z^{n+1} \cdot \frac{n!}{5^{n} } \frac{1}{z^n}<1 \\ \frac{5}{(n+1)} z<1 \\  z \frac{5}{n+1}<1 \\ \text{ remember } n \rightarrow \infty \] maybe you are right

freckles
 one year ago
Best ResponseYou've already chosen the best response.0like because there we would get z*0<1 which is true 0<1 for all z

freckles
 one year ago
Best ResponseYou've already chosen the best response.0oh so nevermind I think it works

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1But it only shows that it is absolutely convergent but not necessarily analytic over the Reals right?

freckles
 one year ago
Best ResponseYou've already chosen the best response.0I don't know about all of that. I just know it shows it converges for all z since we have z*0<1 is true for all z therefore the radius of convergence is infinite there is a similar example here: http://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx in example 4....

freckles
 one year ago
Best ResponseYou've already chosen the best response.0they do use root test there though instead of ratio test

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Unfortunately, my Prof doesn't accept ratio test. We Must do Hamadard test.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1\(limsup \sqrt[n]\dfrac{5^n}{n!} = limsup \dfrac{5}{n/e} = limsup \dfrac{5e}{n} = \infty \) Hence \(R = \dfrac {1}{\infty}= ?\) I don't know how to argue then, since it is undefined.

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1\[ \limsup_{n\to\infty} \sqrt[n]{\dfrac{5^n}{n!}} = \limsup_{n\to\infty} \dfrac{5}{n/e} = \limsup_{n\to\infty} \dfrac{5e}{n} = 0 \] Since n is to infinity right?

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1I am not so sure about dropping the \(\sqrt{2\pi n}\) part of the Stirling's approximation.

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1But then how to argue for R?

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1R is a number, not limit anymore. 1/0 is undefined

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Can we say the radius of convergence is infinitive?

thomas5267
 one year ago
Best ResponseYou've already chosen the best response.1If the sequence values are unbounded so that the lim sup is ∞, then the power series does not converge near a, while if the lim sup is 0 then the radius of convergence is ∞, meaning that the series converges on the entire plane. https://en.wikipedia.org/wiki/Cauchy%E2%80%93Hadamard_theorem

Loser66
 one year ago
Best ResponseYou've already chosen the best response.1Yes, I got you. I know that the radius of convergence is infinitive is different from divergence.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.