Math Is Fun Forum / Convergence

Johnathon bresly · 2013-02-09 22:49:33

What is the difference between absolute and conditional convergence?[examples will be appreciated]

bobbym · 2013-02-10 03:43:18

Hi;

Here is a nice video:

http://www.youtube.com/watch?v=6hOeqjoQvNw

scientia · 2013-02-12 00:39:33

The sequence $http://latex.codecogs.com/gif.latex?\sum_na_n$ is absolutely convergent iff both $http://latex.codecogs.com/gif.latex?\sum_{n=0}^{\infty}a_n$ and $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty|a_n|$ converge.

It is conditionally convergent iff $http://latex.codecogs.com/gif.latex?\sum_{n=0}^{\infty}a_n$ converges while $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty|a_n|$ diverges.

Examples.

$http://latex.codecogs.com/gif.latex?\sum_n\frac{(-1)^n}{2^n}$ is absolutely convergent. We have $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty\frac{(-1)^n}{2^n}=1-\frac12+\frac14-\frac18+\cdots=\frac23$ and $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty\left|\frac{(-1)^n}{2^n}\right|=1+\frac12+\frac14+\cdots=2$ .

$http://latex.codecogs.com/gif.latex?\sum_n\frac{(-1)^n}n$ is conditionally convergent. We have $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty\frac{(-1)^n}n=1-\frac12+\frac13-\frac14+\cdots=\ln2$ while $http://latex.codecogs.com/gif.latex?\sum_{n=0}^\infty\left|\frac{(-1)^n}n\right|=1+\frac12+\frac13+\cdots$ is divergent.