example of ratio test

Consider the sequence given by $a_{n}=x^{n}$ (geometric progression) where $|x|<1$ . Then the series

$\sum_{j=0}^{\infty}a_{n}$

converges. To see this, we can use the ratio test. We need to consider the sequence $|a_{n+1}/a_{n}|$ . But for any $n\geq 0$ we have (when $x\neq 0$ )

$\left|\frac{a_{n+1}}{a_{n}}\right|=\left|\frac{x^{n+1}}{x^{n}}\right|=|x|<1,$

and therefore the series converges. The ratio test and the previous argument shows that the geometric series diverges for $|x|>1$ .

Title	example of ratio test
Canonical name	ExampleOfRatioTest
Date of creation	2013-03-22 15:03:20
Last modified on	2013-03-22 15:03:20
Owner	drini (3)
Last modified by	drini (3)
Numerical id	9
Author	drini (3)
Entry type	Example
Classification	msc 26A06
Classification	msc 40A05