Power Series Question

Hello, thehollow89!

The algebra of the Ratio Test is very tricky . . .

Quote:

$\sum^{\infty}_{n=1} n^nx^n$

We have: . $R \;=\;\frac{a_{n+1}}{a_n} \;=\;\frac{(n+1)^{n+1}x^{n+1}} {n^nx^n} \;=\;\frac{(n+1)(n+1)^n}{n^n}\,x<br />$

. . . . . . . . . $= \;(n+1)\left(\frac{n+1}{n}\right)^n\!\!x \;=\;(n+1)\left(1 + \frac{1}{n}\right)^n\!\!x$

$\text{Take the limit: }\:\lim_{n\to\infty}\left[(n+1)\left(1 + \frac{1}{n}\right)^n x\right]$

. . . . . . . . $=\;\underbrace{\left[\lim_{n\to\infty}(n+1)\right]}_{\text{This is }\infty}\cdot\underbrace{\left[\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n\,\right]}_{\text{This is }e}\cdot x$

. . . . . . . . $=\;\; \infty\cdot e\cdot x \;\;=\;\;\infty\quad\hdots\quad\text{The series diverges.}$