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To find the radius of convergence of <math>\sum_{n=0}^\infty (n!)z^{n!}</math>, you'll need to use the Ratio Test. | To find the radius of convergence of <math>\sum_{n=0}^\infty (n!)z^{n!}</math>, you'll need to use the Ratio Test. | ||
| − | <math>\frac{u_{n+1 | + | <math>\frac{u_{n+1)}{u_n}=\frac{(n+1)!z^{(n+1)!}}{n!z^{n!}}=(n+1)z^{n\cdot n!</math>. |
Ask yourself what that does as n goes to infinity in case |z|<1, =1, >1. You'll need to recall that <math>nr^n\to 0</math> as <math>n\to\infty</math> if <math>|r|<1</math>. | Ask yourself what that does as n goes to infinity in case |z|<1, =1, >1. You'll need to recall that <math>nr^n\to 0</math> as <math>n\to\infty</math> if <math>|r|<1</math>. | ||
Revision as of 06:53, 13 November 2009
Discussion area to prepare for Exam 2
To find the radius of convergence of $ \sum_{n=0}^\infty (n!)z^{n!} $, you'll need to use the Ratio Test.
$ \frac{u_{n+1)}{u_n}=\frac{(n+1)!z^{(n+1)!}}{n!z^{n!}}=(n+1)z^{n\cdot n! $.
Ask yourself what that does as n goes to infinity in case |z|<1, =1, >1. You'll need to recall that $ nr^n\to 0 $ as $ n\to\infty $ if $ |r|<1 $.
