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Topic review (newest first)

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.

bobbym
2013-02-10 03:43:18
Johnathon bresly
2013-02-09 22:49:33

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

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