Worked out examples using Cauchy criterion for series

In summary, the conversation discusses the Cauchy Criterion for Series and the request for a website with worked out examples. It also mentions the use of grouping terms in 2^n groups to investigate s_{2^n} - s_{2^{n-1}} for the series 1/n and 1/(n(n+1)). The conversation also mentions the use of other convergence tests and provides a simpler case to consider.
  • #1
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Hello everyone,

Can anybody suggest a website that has worked out examples using the Cauchy Criterion for Series? or, if your feeling ambitious, work out the following problems below:

1.
[tex]\sum^{\infty}_{n=1}1/n[/tex]

2.

[tex]\sum^{\infty}_{n=1}1/(n(n+1))[/tex]The reason why I'm asking for this is because our textbook introduces the theorem (Steven Lay) but it does not demonstrate its usage. I have also perused the web and was not able to find a complete demonstration.

I only need to see it being used. The problems above are not homework questions. I promise.

I appreciate your help

M
 
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  • #2
Check here for a review of the test and for reference to the variables I am referring to; http://en.wikipedia.org/wiki/Cauchy's_convergence_test.

Basically the trick is finding suitable values for m and n that make sure s_m - s_n can not be smaller than epsilon. This can be done in many ways but there's no systematic method. One way is to try to compare it to another series. The first example is a well known proof.

1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + ...+ 1/16) + (1/17+...+ 1/32) ... continually grouping them into groups of 2^n terms, you might want to investigate what each group must be larger than.

In terms of Cauchy, you are investigating [itex]s_{2^n} - s_{2^{n-1}}[/itex].
 
  • #3
Thank you for your reply.

What do you mean by grouping them into 2^n groups?
 
  • #4
michonamona said:
Thank you for your reply.

What do you mean by grouping them into 2^n groups?

Think he means groups of (1/2)^n terms

ie

1 + 1/2 + (1/3 + 1/4) + (1/5 + 1/6 + 1/7 + 1/8) + (1/9 + ...+ 1/16) + (1/17+...+ 1/32) ...

>

1 + 1/2 + (1/4 + 1/4) + (1/8 + 1/8 + 1/8 + 1/8) + (1/16 + ...+ 1/16) + (1/32+...+ 1/32) ...

And note that s2n-1 = n+1 for the second sequence of partial sums

Then think about convergence tests you know
 
  • #5
Being a fool I made it harder than it had to be! Simpler case to look at is m=2n.
 

1. What is the Cauchy criterion for series?

The Cauchy criterion for series states that a series converges if and only if for any positive number ε, there exists a positive integer N such that for all n > N and m > N, the absolute value of the difference between the sum of the terms from n+1 to m is less than ε.

2. How is the Cauchy criterion used to determine convergence of a series?

The Cauchy criterion is used to determine convergence of a series by checking if the conditions stated in the criterion are satisfied. If the conditions are satisfied, then the series is said to be convergent. If not, then the series is said to be divergent.

3. What is the significance of the Cauchy criterion in mathematics?

The Cauchy criterion is significant in mathematics because it provides a precise and rigorous way to determine the convergence of a series. It also allows for the comparison of different series and helps establish relationships between different types of series.

4. Can the Cauchy criterion be used for all types of series?

Yes, the Cauchy criterion can be used for all types of series, as long as the terms in the series are real or complex numbers. It can also be used for both infinite and finite series.

5. How does the Cauchy criterion differ from other convergence tests for series?

The Cauchy criterion differs from other convergence tests for series in that it focuses on the behavior of the partial sums of a series, rather than the individual terms. It also provides a more general and powerful method for determining convergence, as it can be applied to a wide range of series.

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