adri

I seem to have some trouble with understanding this criterion fully...

So what I think that I understand:

No matter how small I take my epsilon, I will always find a smaller number (if the series is convergent) by subtracting two numbers of a row.

For example:

Row (goes to the square root of 2): 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ...

Let's take epsilon = 1/100.000

Then I can always find a number smaller than epsilon:

1.414213 - 1.41421 = 3*10^-6 < epsilon

And if epsilon is even smaller, you just go further down the row to get 2 numbers that are even closer to each other. It makes sense to me because if you go really far in the row, the distance between two numbers is getting smaller and smaller until it reaches 0 at infinity which means there is a number and therefor convergent.

Questions:

Thanks for your answer.

adri

Cauchy wrote: |

Let (an) be a sequence [R or C]. We say that (an) is a Cauchy sequence if, for all ε > 0 there exists N ∈ N such that
m, >= N =⇒ |am − an| < ε. |

So what I think that I understand:

No matter how small I take my epsilon, I will always find a smaller number (if the series is convergent) by subtracting two numbers of a row.

For example:

Row (goes to the square root of 2): 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, ...

Let's take epsilon = 1/100.000

Then I can always find a number smaller than epsilon:

1.414213 - 1.41421 = 3*10^-6 < epsilon

And if epsilon is even smaller, you just go further down the row to get 2 numbers that are even closer to each other. It makes sense to me because if you go really far in the row, the distance between two numbers is getting smaller and smaller until it reaches 0 at infinity which means there is a number and therefor convergent.

Questions:

- Are my thoughts (semi-)correct?

- What is N? I don't see it in the actual formula?

- Can someone fill in the formula with my example (so what's am, an, m, n and N in the formula?) and work it out, because on every website I visited they just take a row and in one line they say it is convergent without a numerical solution.

Thanks for your answer.

adri