Another Test for Convergence Question

In summary, the conversation discusses how to test the convergence or divergence of the series $$\sum_{n=1}^{\infty} \frac{1}{3^{\ln n}}$$ The participants suggest using the laws of exponentials and rewriting the series as a power series using the fact that ##3=e^{\ln 3}##. It is concluded that the series is convergent due to the p-test.
  • #1
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4

Homework Statement


Test the following series for convergence or divergence.
$$\sum_{n=1}^{\infty} \frac{1}{3^{\ln n}}$$

The Attempt at a Solution


I've tried to compare this to geometric series ##3^n## but obviously the target term is larger overall than its geometric counterpart. Comparing with the reciprocal of ##\ln n## also brings no avail because the term is smaller than the former. Ratio test gives ##ρ=1## which is inconclusive. It certainly passes preliminary test because the limit goes to 0 once n goes to infinity. I haven't done integral test because it's quite ugly and I don't know how to integrate the corresponding function.

Can you guys give me suggestion to tame this series down nicely?

Thank You
 
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  • #2
Use that 3=e^(ln(3)) and use the laws of exponentials to rearrange it a bit. It becomes a power series.
 
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  • #3
Edit: Dick beat me to it. Quick.
 
Last edited:
  • #4
Try using 3 =eln(3) to rewrite your summand
 
  • #5
Strange things happened when I arrange the summation.

##3=e^{\ln 3}##
$$3^{\ln n}=(e^{\ln 3})^{\ln n}$$
$$3^{\ln n}=e^{\ln 3 ° \ln n}$$
$$3^{\ln n}=n^{\ln 3}$$

If that's the case then the series must be convergent because of p-test, with p being larger than 1.
 
  • #6
Seydlitz said:
Strange things happened when I arrange the summation.

##3=e^{\ln 3}##
$$3^{\ln n}=(e^{\ln 3})^{\ln n}$$
$$3^{\ln n}=e^{\ln 3 ° \ln n}$$
$$3^{\ln n}=n^{\ln 3}$$

If that's the case then the series must be convergent because of p-test, with p being larger than 1.

I don't know if I'd call that strange, but it is the correct conclusion.
 
  • #7
Dick said:
I don't know if I'd call that strange, but it is the correct conclusion.

Well maybe I'm not quite used to manipulating exponents yet. Thanks for your help guys!
 

1. What is convergence?

Convergence is the process by which a series approaches a finite limit as more terms are added.

2. How do you test for convergence?

There are various methods for testing convergence, such as the comparison test, the ratio test, and the root test.

3. What is the purpose of testing for convergence?

The purpose of testing for convergence is to determine whether a series will have a finite sum or will continue to increase indefinitely. This can help in understanding the behavior of a series and its potential applications.

4. Can a series have a finite limit but still not converge?

Yes, a series can have a finite limit but still not converge. This is known as conditionally convergent, where the series alternates between positive and negative terms and does not approach a specific value.

5. How does the divergence test work?

The divergence test, also known as the nth term test, states that if the limit of the nth term of a series is not equal to zero, then the series diverges. In other words, if the individual terms of the series do not approach zero, then the entire series will not converge.

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