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

Johnathon bresly
Guest

### Convergence

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

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

bobbym
From: Bumpkinland
Registered: 2009-04-12
Posts: 91,517

### Re: Convergence

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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

scientia
Member
Registered: 2009-11-13
Posts: 222

### 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.

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