- #1
LukeMiller86
- 5
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1. The problem statement:
Show that the following series has a radius of convergence equal to [tex] exp\left(-p\right) [/tex]
For p real:
[tex]\Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n}[/tex]
[tex]\stackrel{lim}{n\rightarrow\infty}\left|a_{n}\right|^{1/n} = \frac{1}{R} = \left(\frac{n+p}{n}\right)^{n}
=exp\left(n\left(ln\left(\frac{n+p}{n}\right)\right)\right)[/tex]
Apart from playing with the logarithm after that I cannot seem to reach the required answer.
Any help would be greatly appreciated.
Show that the following series has a radius of convergence equal to [tex] exp\left(-p\right) [/tex]
Homework Equations
For p real:
[tex]\Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n}[/tex]
The Attempt at a Solution
[tex]\stackrel{lim}{n\rightarrow\infty}\left|a_{n}\right|^{1/n} = \frac{1}{R} = \left(\frac{n+p}{n}\right)^{n}
=exp\left(n\left(ln\left(\frac{n+p}{n}\right)\right)\right)[/tex]
Apart from playing with the logarithm after that I cannot seem to reach the required answer.
Any help would be greatly appreciated.