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#1 2013-02-09 22:49:33

Johnathon bresly
Guest

Convergence

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

#2 2013-02-10 03:43:18

bobbym

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Re: Convergence

In mathematics, you don't understand things. You just get used to them.
Some cause happiness wherever they go; others, whenever they go.
If you can not overcome with talent...overcome with effort.

#3 2013-02-12 00:39:33

scientia
Full Member

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Re: Convergence

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

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

Examples.

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

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