Radius of Convergence for Moderately Complicated Series

[LukeMiller86]

1. The problem statement:

Show that the following series has a radius of convergence equal to $$exp\left(-p\right)$$

Homework Equations

For p real:

$$\Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n}$$

The Attempt at a Solution

$$\stackrel{lim}{n\rightarrow\infty}\left|a_{n}\right|^{1/n} = \frac{1}{R} = \left(\frac{n+p}{n}\right)^{n}<br /> =exp\left(n\left(ln\left(\frac{n+p}{n}\right)\right)\right)$$

Apart from playing with the logarithm after that I cannot seem to reach the required answer.
Any help would be greatly appreciated.

 

[foxjwill]

What's the limit definition of the exponential function?

 

[LukeMiller86]

$$exp\left(-p\right) = e^{\left(-p)\right}$$

is that what you meant?

 

[g_edgar]

Do you know this limit:
$$\lim_{n\to\infty}\left(1+\frac{p}{n}\right)^n$$

 

[LukeMiller86]

Completely overlooked that! Thanks very much.