Radius of Convergence for Moderately Complicated Series

1. Sep 6, 2009

LukeMiller86

1. The problem statement:

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

2. Relevant equations

For p real:

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

3. The attempt at a solution
$$\stackrel{lim}{n\rightarrow\infty}\left|a_{n}\right|^{1/n} = \frac{1}{R} = \left(\frac{n+p}{n}\right)^{n} =exp\left(n\left(ln\left(\frac{n+p}{n}\right)\right)\right)$$

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

2. Sep 6, 2009

foxjwill

What's the limit definition of the exponential function?

3. Sep 6, 2009

LukeMiller86

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

is that what you meant?

4. Sep 6, 2009

g_edgar

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

5. Sep 6, 2009

LukeMiller86

Completely overlooked that! Thanks very much.