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

  • Thread starter catsfanj
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In summary, the conversation discusses a problem involving the summation of a series and the use of the comparison test to prove its convergence. One person suggests writing the series as Ʃk1/k-3 and finding a real number that makes the series convergent. The other person asks for help in determining the convergence of the series \sum_k k^r.
  • #1
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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  • #2
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?
 
  • #3
No I can't.
 
  • #4
Can you tell us for which real numbers r, the series [itex]\sum_k k^r[/itex] converges?
 
  • #5
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.
 
  • #6
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?
 

1. What is the purpose of the Comparison Test in solving infinite series?

The Comparison Test is used to determine whether an infinite series converges or diverges by comparing it to another known series with known convergence or divergence.

2. How does the Comparison Test work?

The Comparison Test works by comparing the terms of the given series to the terms of a known series with known convergence or divergence. If the known series converges and the given series has terms that are smaller or equal to the terms of the known series, then the given series also converges. If the known series diverges and the given series has terms that are larger or equal to the terms of the known series, then the given series also diverges.

3. Can the Comparison Test be used for all infinite series?

No, the Comparison Test can only be used for series with positive terms. It also cannot be used for series where the terms alternate in sign.

4. What is the role of the kth root of k in the infinite series being solved?

The kth root of k is the term used to represent the general term of the infinite series being solved. It is raised to the power of 1/k^3, which is the power of the denominator in the given series. This term is used in the Comparison Test to compare to the known series.

5. How do you determine whether the given series converges or diverges using the Comparison Test?

If the known series converges and the given series has terms that are smaller or equal to the terms of the known series, then the given series also converges. If the known series diverges and the given series has terms that are larger or equal to the terms of the known series, then the given series also diverges. If the Comparison Test cannot be applied, other tests such as the Ratio Test or the Integral Test can be used to determine convergence or divergence.

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