Diverging and converging infinite series

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Discussion Overview

The discussion revolves around the convergence and divergence of infinite series, particularly focusing on the application of various convergence tests and methods for determining the behavior of specific series. Participants explore both theoretical aspects and practical calculations related to series convergence.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant discusses applying the ratio test to their series and expresses uncertainty about the method, particularly regarding the limit test and whether to divide the numerator by n.
  • Another participant suggests using the integral test for convergence as an alternative approach.
  • A different participant notes that the terms in the sum behave like 1/n for large n, indicating that the series is well-known to be divergent and behaves like ln(n).
  • A participant shares their experience with a calculator and describes a process that leads to a convergence result of 4, questioning the correctness of their method and seeking clarification on forming an approximation of a function using Newton's method.
  • One participant clarifies that their previous expression refers to the term at infinity rather than summation to infinity.

Areas of Agreement / Disagreement

Participants express differing views on the methods for determining convergence, with some proposing specific tests while others question the validity of their approaches. The discussion remains unresolved regarding the correctness of the methods and results presented.

Contextual Notes

There are limitations in the clarity of the methods discussed, particularly regarding the application of the ratio test and the integral test. Some assumptions about the series behavior and convergence criteria are not fully articulated, leaving room for interpretation.

Who May Find This Useful

This discussion may be useful for students and enthusiasts of mathematics who are exploring the concepts of infinite series, convergence tests, and numerical methods for approximating functions.

eptheta
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I looked through some tutorials to find intervals of divergence and tests for divergence...
My series:
[PLAIN]http://img843.imageshack.us/img843/4193/51453212.jpg
a and x are constants...
I did the ratio test and i get [tex]\rho[/tex]=1, so i tried to apply the limit test to see if an is zero or does not exist... The problem is, i am not sure about my method...
Do i divide the numerator(1) by n also, if so, the test gives an=0... But then again i am not sure...
I learned about these tests from http://www.math.unh.edu/~jjp/radius/radius.html"...

Could someone please tell me if this sum is convergent or divergent (and the method you used to find this out)

Thanks
 
Last edited by a moderator:
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Take x outside the summation. The terms in the sum behave like 1/n for large n. This series is well known divergent. The sum behaves like ln(n).
 
That makes sense, thanks.. I have one more question...

I was fiddling with my calculator and typed "Ans-(Ans-[tex]\sqrt{Ans}[/tex]-2)" and kept clicking =...(The expression is in the same form as xn - f(x)/f'(x) but is just equal to [tex]\sqrt{Ans}[/tex]+2...
I found that it converges as 4. It seemed logical enough...But then i decided to form the equation that represents f(x)...

so basically, what i got out of the expression is that y/(dy/dx)=x-[tex]\sqrt{x}[/tex]-2
dy/y=dx/(x-[tex]\sqrt{x}[/tex]-2)
Integrating(painfully), ln y=k(x-[tex]\sqrt{x}[/tex]-2)([tex]\sqrt{x}[/tex]-2/[tex]\sqrt{x}[/tex]+2)1/3

Its root x -2 in the numerator not root 3 -2...

Where k is a constant... I should have tried with something simpler, but is this the correct way to form an approximation of a function that converges using Newton's method (or am i just being an oaf)
If it is the former, then how do i find k.
If it is the latter, then how do i find f(x) for :
[PLAIN]http://img714.imageshack.us/img714/5351/96407726.png

Thanks.
 
Last edited by a moderator:
Sorry, it is not summation to infinity, but rather term at infinity for the last expression.
-Thanks
 

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