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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: 85,270

### Re: Convergence

In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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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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