Series convergence and Cauchy criterion

Thread starter estro
Start date May 29, 2010
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Cauchy Convergence Series Series convergence

In summary, series convergence is the property of a series to have a finite limit as more terms are added, and the Cauchy criterion is a method used to determine if a series converges by showing that the terms become arbitrarily close to zero. The Cauchy criterion is related to the Bolzano-Weierstrass theorem, which states that any bounded sequence contains a convergent subsequence. There are two types of convergence, absolute and conditional, and the Cauchy criterion can be used to determine if a series is convergent by showing that the difference between terms becomes smaller and smaller.

May 29, 2010

estro

The Attempt at a Solution

* forgot to state that I choose m > n > max { N_1, N_2 }.

I'm not sure if i did it right, but seems ok to me =)

Will appreciate your opinion...

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Jun 1, 2010

estro

still struggling with this one...=(

Related to Series convergence and Cauchy criterion

1. What is series convergence?

Series convergence is the property of a series, or an infinite sum of numbers, to have a finite limit. In other words, as you add more and more terms to the series, the sum approaches a specific value. If this value exists, the series is said to converge. If the limit does not exist, the series is said to diverge.

2. What is the Cauchy criterion for series convergence?

The Cauchy criterion is a method used to determine if a series converges. It states that for a series to converge, the terms in the series must become arbitrarily close to zero as the series progresses. In other words, if the difference between any two consecutive terms in the series becomes smaller and smaller, the series is said to converge.

3. How is the Cauchy criterion related to the Bolzano-Weierstrass theorem?

The Bolzano-Weierstrass theorem is a fundamental theorem in real analysis that states that any bounded sequence contains a convergent subsequence. This theorem is closely related to the Cauchy criterion for series convergence because it can be used to prove that a series converges by showing that it is a bounded sequence and therefore contains a convergent subsequence.

4. What is the difference between absolute convergence and conditional convergence?

A series is said to be absolutely convergent if the series of the absolute values of its terms converges. This means that the series will converge regardless of the order in which the terms are added. On the other hand, a series is said to be conditionally convergent if it converges, but the series of the absolute values of its terms does not converge. This means that the order of the terms in the series affects its convergence.

5. How can I use the Cauchy criterion to determine if a series is convergent?

To use the Cauchy criterion, you must first show that the series satisfies the Cauchy criterion. This means that you must show that the difference between any two consecutive terms in the series becomes smaller and smaller as the series progresses. If this is true, then the series is said to converge. However, if you cannot prove that the series satisfies the Cauchy criterion, you cannot determine if the series converges using this method.