Convergence of ln(k)/k^3 Series

  • Thread starter zeion
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But that doesn't guarantee that ln(x) < x. You'll have to use the definition of the natural logarithm to prove that.
  • #1
zeion
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Homework Statement



Determine whether the series converges or diverges.

[tex] \sum \frac{lnk}{k^3} [/tex]

Homework Equations


The Attempt at a Solution



Since lnk always less than 0, so [tex] \frac{lnk}{k^3} \leq \frac{1}{k^3}[/tex] and [tex]\frac{1}{k^3} [/tex]diverges so[tex] \frac{lnk}{k^3}[/tex] diverges.
 
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  • #2
zeion said:

Homework Statement



Determine whether the series converges or diverges.

[tex] \sum \frac{lnk}{k^3} [/tex]


Homework Equations





The Attempt at a Solution



Since lnk always less than 0, so [tex] \frac{lnk}{k^3} \leq \frac{1}{k^3}[/tex] and [tex]\frac{1}{k^3} [/tex]diverges so[tex] \frac{lnk}{k^3}[/tex] diverges.
I count three mistakes here.
1. Why do you think that ln(k) is always < 0? I'm assuming that k = 1, 2, 3, ...
2. The series whose general term is 1/k3 converges.
3. If you want to show that a given series diverges, its terms must be larger than those of a divergent series. If the terms of the given series are smaller than those of a divergent series, you can't conclude anything.
 
  • #3
Can I argue that [tex]ln(k) < k [/tex] then [tex] \frac{ln(k)}{k^3} < \frac{k}{k^3} [/tex]then it converge because [tex]\frac{1}{k^2} [/tex] converges?
 
  • #4
I can buy that. However, your should be able to convince yourself that ln(x) < x.
 
  • #5
Yeah since ln(x) is the inverse of e^x and they reflect on the y =x.
 
  • #6
The fact that y = ln(x) and y = ex are reflections across the line y = x doesn't prove that ln(x) < x. If y = ex happens to cross this line a few times means that ln(x) will do so, also.
 

Related to Convergence of ln(k)/k^3 Series

1. What does it mean for a series to converge or diverge?

A series is said to converge if the sequence of partial sums approaches a finite limit as the number of terms increases. In other words, as more terms are added to the series, the overall sum gets closer and closer to a certain value. On the other hand, a series is said to diverge if the sequence of partial sums does not approach a finite limit, meaning that the overall sum does not have a defined value.

2. How do you determine if a series converges or diverges?

There are several tests that can be used to determine if a series converges or diverges. These include the ratio test, comparison test, integral test, and limit comparison test. These tests involve evaluating the behavior of the terms in the series to determine if they approach zero or a finite limit. The results of these tests can indicate whether the series converges or diverges.

3. Can a series converge and diverge at the same time?

No, a series can only either converge or diverge, it cannot do both. If a series converges, it means that the overall sum approaches a finite value. If it diverges, it means that the overall sum does not have a defined value. These two outcomes are mutually exclusive.

4. What is an alternating series and how does it converge?

An alternating series is a series in which the terms alternate between positive and negative values. These types of series can converge even if the terms do not approach a finite limit. In an alternating series, the terms must decrease in magnitude and approach zero in order for the series to converge. This is known as the alternating series test.

5. Can a series converge if its terms do not approach zero?

No, if the terms of a series do not approach zero, then the series will not converge. This is because the terms are an essential factor in determining the behavior of the series. If the terms do not approach zero, then the series will either diverge or have an undefined limit.

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