Convergence

Johnathon bresly · 2013-02-08 23:49:33

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

bobbym · 2013-02-09 04:43:18

Hi;

Here is a nice video:

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

scientia · 2013-02-11 01:39:33

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

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

Examples.

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

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

Math Is Fun Forum

#1 2013-02-08 23:49:33

Convergence

#2 2013-02-09 04:43:18

Re: Convergence

#3 2013-02-11 01:39:33

Re: Convergence

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