# Thread: Ratio and Root Test

1. ## Ratio and Root Test

How would you work out if these series converge or diverge?

Problem 1

n=1 to infinity n!/10^n

Problem 2

n=1 to infinity ((n-2)/n)^n

Problem 3

n=1 to infinity (-2)^n/3^n

2. What exactly are you having troubles with? You are already told which tests to use.

For example, the first one. Let $\displaystyle a_n = \frac{n!}{10^n}$.

Then: $\displaystyle \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| = \lim_{n \to \infty} \left| \frac{(n+1)!}{10^{n+1}} \cdot \frac{10^n}{n!}\right| = \lim_{n \to \infty} \left|\frac{(n+1)n!}{10^n \cdot 10} \cdot \frac{10^n}{n!}\right| = \cdots$

3. still not sure how to do the third problem though....can't use root test because of the negative term. There too many tests....

4. Originally Posted by TAG16
still not sure how to do the third problem though....can't use root test because of the negative term. There too many tests....
Problem 3

n=1 to infinity (-2)^n/3^n
$\displaystyle \sum_{n=1}^{\infty} \frac{(-2)^n}{3^n} = \sum_{n=1}^{\infty} \left(\frac{-2}{3}\right)^n$

geometric series with $\displaystyle |r| < 1$ ???