Help with convergence/divergence

  • Thread starter Song
  • Start date
In summary, the series given by Sigma (-1)^n / (ln n)^n is conditionally convergent, as it satisfies the alternating series test but fails the ratio test. The first series, Sigma 1/(n*ln(n)), is also convergent as it satisfies the integral test. However, the second series, Sigma (-1)^(n-1)/n^p, is only convergent for p>0 as p=0 would fail the alternating series test. The last series, Sigma Csubn, can be said to be convergent for (a) and (c) while divergent for (b) and (d). The integral 2 to infinity lnx/x^2 can be
  • #1
Song
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Sigma (-1)^n / (ln n)^n

First I start using the ratio test (because I want to test if it's absolutely conver. or conditionally conver.), then I get the limit = 1 so it's inconclusive by the ratio test.
But from the alternating series test, it's convergent. So it's conditionally convergent.

Is my process right?
 
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  • #2
limit absolute value of nln(n)/ (n+1)(ln(n+1))
--->n/(n+1) so the limit is 1.
 
  • #3
yes you are correct. it is conditionally convergent, because it satisfies the alternating series test but fails the ratio test.
 
  • #4
As you've written it in the first post, [itex](\ln(n))^n\neq n \ln(n)[/itex], it's [itex]\ln(n^n)= n \ln(n)[/itex]. Which did you mean?
 
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  • #5
sigma n=1 to infinite 1/(n*ln(n))
sigma n=1 to infinite (-1)^n /(n*ln(n))

The second one is convergent, how about the first one? It's convergent as well?
 
  • #6
To StatusX,
Find the limit for absolute value of nln(n)/ (n+1)(ln(n+1))
I cross out ln(n) on the top with the ln(n+1) at the bottom, so it left with n/(n+1). Do you mean this one?
For the first one, ln(n)^n is at bottom. So it's same as nln(n).
 
  • #7
[tex] \sum_{n=1}^{\infty} \frac{1}{n\ln n} [/tex].

Use the integral test.[tex] \lim_{x\rightarrow \infty}\frac{n\ln n}{(n+1)(\ln(n+1))} = 1 [/tex]
 
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  • #8
I see. It's divergent for the second one but is convergent for the first one because of Alternating series test.

Thanks a lot.
 
  • #9
Here's another question.

Sigma n=1 to infinity (-1)^(n-1)/n^p

For what values of p is serie convergent?

I have p>0, can p be 0 in this case?
 
  • #10
I'm saying there's a difference between [itex](\ln(n))^n[/itex] and [itex]\ln(n^n)[/itex], and the one you've written in you're first post is not equal to [itex]n \ln(n) [/itex].
 
  • #11
[tex] \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^{p}} [/tex]. p can't be 0, because then [tex] a_{n} [/tex] wouldn't be decreasing, and it would fail the alternating series test. Thus [tex] p>0 [/tex].
 
  • #12
To StatusX,
I see what you mean now.
Let me re-do my problem.
Thanks by the way.
 
  • #13
For [tex] \sum_{n=1}^{\infty}\frac{(-1)^{n} }{(\ln n)^{n}} [/tex], you could use the root test. So it is absolutely convergent.
 
  • #14
Thank you.
Here's a question I don't understand.

Suppose that sigma n=0 to the infinity Csubn*x^n converges when x=-4 and diverges when x=6. What can be said about the convergence or divergence of the following series?

a) Sigma n=0 to infinity Csubn
.....

Can someone give me a hint on this one?
 
  • #16
Thanks. But I'm not sure if I get the concept.
 
  • #17
a)con.
b)div.
c)con.
d)div.
 
  • #18
yeah that's right
 
  • #19
I sort of get the concept...but not exactly sure. Thank you so much though.
 
  • #21
I don't think so, because [tex] a_{n} [/tex] is not decreasing.
 
  • #22
Yeah. That's right.
How do you do the integral 2 to infinity lnx/x^2?
Integration by parts?
 
  • #23
yeah int by parts
 

What is convergence and divergence in scientific terms?

Convergence and divergence refer to the behavior of a sequence or series, where convergence means that the terms of the sequence or series approach a finite limit as the number of terms increases, and divergence means that the terms do not approach a finite limit.

What is the importance of understanding convergence and divergence?

Understanding convergence and divergence is important in many scientific fields, as it allows for accurate predictions and interpretations of data. It is particularly relevant in fields such as mathematics, physics, and engineering, where it is used to analyze and model complex systems.

How can convergence and divergence be tested?

Convergence and divergence can be tested using various methods, such as the ratio test, the comparison test, and the integral test. These methods involve comparing the given sequence or series to a known convergent or divergent sequence or series.

What are some real-life examples of convergence and divergence?

An example of convergence in real life is the decreasing population of a species as resources become limited. An example of divergence is the spread of a virus in a population, where the number of infected individuals increases exponentially.

What are the implications of convergence and divergence in scientific research?

The implications of convergence and divergence in scientific research are vast, as it allows for the analysis and understanding of complex systems and phenomena. It also helps in making accurate predictions and identifying patterns in data, leading to advancements in various fields of science.

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