Converge or Diverge: Solving ∞Ʃ (kth root of k)/k^3 using Comparison Test

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Homework Help Overview

The problem involves determining the convergence of the series ∞Ʃ (kth root of k)/k^3 using the comparison test. The subject area pertains to series convergence and comparison tests in calculus.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the application of the comparison test and express uncertainty about how to proceed. There are attempts to reformulate the series and questions about the convergence criteria for related series.

Discussion Status

The discussion is ongoing, with participants exploring different aspects of the problem. Some have suggested rewriting the series and are inquiring about the conditions under which certain series converge, indicating a productive exchange of ideas.

Contextual Notes

There is a lack of clarity regarding the specific series that should be used for comparison, and participants are questioning the assumptions related to convergence criteria.

catsfanj
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Homework Statement


Ok, so I was given this problem:

Ʃ (kth root of k)/k^3 = k^(1/k) / k^3
k=1

Homework Equations


None


The Attempt at a Solution


I know that I have to use the comparison test, but am unsure how to apply it. Can somebody help me?
 
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You can write that into Ʃk1/k-3. Can you find a nice real number r that has the property that r>1/k-3 for all k and Ʃkr converges?
 
No I can't.
 
Can you tell us for which real numbers r, the series [itex]\sum_k k^r[/itex] converges?
 
I know that the series converges, but do not know what to use for the second series. I need to prove its convergence using one of the tests.
 
catsfanj said:
I know that the series converges, but do not know what to use for the second series. I need to prove its convergence using one of the tests.

What series converges?? What second series?? :confused:

Can you answer my question about [itex]\sum k^r[/itex]?? For which r does it converge?
 

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