Not sure how to proceed with this series

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In summary, the conversation discusses various methods for determining the convergence or divergence of different series. The main topics covered are the alternating series test, using L'Hopital's rule, and the sum of geometric series. The conversation also includes questions about specific series, such as \sum_{n= 1}^\infty \frac{n}{ln n} and \sum_{n= 1}^\infty \frac{6^n + 5^n}{30^n}. The conclusion is that \frac{n}{ln n} diverges and \frac{6^n + 5^n}{30^n} can be solved by splitting it into two smaller series and using the sum of geometric series formula.
  • #1
frasifrasi
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With this series,

Summation from 2 to infinity of

(-1)^(n).n/ln(n)

--> Am I supposed to do the alternating series test?

If I do, lim n --> infinity n/ln(n): am I supposed to do l'hopital here or is there a more straight foward method?


Thank you.
 
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  • #2
L'hopital is a pretty good way to do [tex]\frac {n}{ln(n)}[/tex] as n goes to infinity
 
  • #3
ok, it goes to infinity, but the first part of the alternating series, how can I determined that [tex]\frac {n+1}{ln(n+1)}[/tex]?

also, for a different series altogether, I am not sure how [tex]\frac {(-1 to (n-1))}{2n}[/tex] --> -1^(n-1)
is conditionally convergent. If I take the abs value, I get 1/2n, which is a p series and diverges. And the alternating series diverges, so would it not diverge?
 
Last edited:
  • #4
To be completely honest, I have no idea what you're trying to ask. Did you mess up the latex?

The fact that the limit goes to infinity specifically means it doesn't go to zero, which is a necessary condition for convergence
 
  • #5
but the first series I posted is supposed to be conditionally convergent. Can anyone else help?
 
  • #6
frasifrasi said:
but the first series I posted is supposed to be conditionally convergent.

It isn't.
 
  • #7
That is what it says in the answer key, condinionally...

Can anyone else comment?
 
  • #8
frasifrasi said:
That is what it says in the answer key, condinionally...

Can anyone else comment?

Divergent. The key is simply wrong or the problem is misstated.
 
  • #9
Are you talking about the same series?

The "last series" referred to, I think, was
[tex]\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}[/tex]
an "alternating series". That converges if and only if the associated sequence converges to 0 which is true. Of course, the series of absolute values is
[tex]\sum_{n=1}^\infty \frac{1}{n}[/tex]
which does not converge so the sum converges "conditionally", not "absolutely".


For the first series
[tex]\sum_{n= 1}^\infty \frac{n}{ln n}[/itex]
Since the associated sequence n/ln n does not go to 0 it cannot converge either conditionally nor absolutely.
 
  • #10
so, [tex]\sum_{n= 1}^\infty \frac{n}{ln n}[/tex] --> the lim n-->infinity =

1/ 1/x = infinity, so it diverges, correct?
 
  • #11
frasifrasi said:
so, [tex]\sum_{n= 1}^\infty \frac{n}{ln n}[/tex] --> the lim n-->infinity =

1/ 1/x = infinity, so it diverges, correct?

This would be better stated as
1/(1/n) -> infinity as n-> infinity, so n/ln(n) -> infinity as n->infinity by L'Hopital. Hence, the sum is divergent, as the summation terms do not tend to zero
 
  • #12
What about [tex]\sum_{n= 1}^\infty \frac{6^n + 5^n}{30^n}[/tex]


- it's been a while, what method should I use for this?
 
  • #13
anyone (I am trying to prepare for a test : ) !)
 
  • #14
Can you do [itex]\sum_{n= 1}^\infty x^n[/itex] for [itex]|x|<1[/itex]?
 
  • #15
frasifrasi said:
What about [tex]\sum_{n= 1}^\infty \frac{6^n + 5^n}{30^n}[/tex]


- it's been a while, what method should I use for this?

Big hint of the day: You can split it into 2 small parts, and solve it easily using the sum of geometric series.

Can you go from here? :)
 

1. What does "Not sure how to proceed with this series" mean?

"Not sure how to proceed with this series" refers to a state of uncertainty or indecision about how to continue or move forward with a particular series of experiments, research, or data analysis.

2. How can I determine the next steps for this series?

The next steps for a series can be determined by carefully evaluating the current data and results, identifying any gaps or unanswered questions, and considering the overall objectives and goals of the research. Consultation with colleagues or mentors can also be helpful in determining the best course of action.

3. What should I do if I am stuck in the middle of a series?

If you find yourself stuck in the middle of a series, it may be helpful to take a step back and reassess the methods and approach being used. It may also be beneficial to seek input from others or try different techniques to overcome any challenges or roadblocks.

4. How do I know if I should continue with this series or move on to a different project?

The decision to continue with a series or move on to a different project should be based on the significance and relevance of the research, the availability of resources and support, and the potential for new insights and discoveries. It is important to carefully weigh the pros and cons before making a decision.

5. Are there any potential risks or limitations to proceeding with this series?

Yes, there can be potential risks or limitations to proceeding with a series, such as technical difficulties, unexpected results, or ethical concerns. It is important to carefully consider and address these potential issues in order to ensure the validity and integrity of the research.

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