Math History: Cauchy Criterion for Sequence/Series

[warwick]

I know the Cauchy criterion for a convergent sequence. After looking at an example in the book, I was able to write this down. It would appear that the sequence converges to zero since the numerator is bounded by -1 and 1. It looks like the Cauchy criterion for convergent series is satisfied too since you can make m and n and a function of epsilon. However, I'm not too sure about my work.

http://i111.photobucket.com/albums/n149/camarolt4z28/Untitled-1.png

http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20111116_173313.jpg

 

[pka]

I know the Cauchy criterion for a convergent sequence.

Here is an intuitive idea of the Cauchy criterion.
If \(\displaystyle (a_n)\) is a sequence having that the property that no matter how small a positive distance is, say \(\displaystyle \epsilon\) from some index value on all of the distance between terms of the sequence is less than \(\displaystyle \epsilon\)

 

[warwick]

Here is an intuitive idea of the Cauchy criterion.
If \(\displaystyle (a_n)\) is a sequence having that the property that no matter how small a positive distance is, say \(\displaystyle \epsilon\) from some index value on all of the distance between terms of the sequence is less than \(\displaystyle \epsilon\)

I'm not quite following your thought. A Cauchy sequence is one in which the distance between successive terms becomes smaller and smaller. You can find a number N such that the terms after that have a a distance that is less than \(\displaystyle \epsilon\).