Convergence of Series: Dyadic Criterion for Series Involving Logs

In summary, the task is to determine the convergence of two series involving logs, and the suggested approach is to use the Dyadic Criterion. However, the equation becomes convoluted and the person is unsure how to proceed. They are seeking guidance on which tests to use and are considering the root test, which ultimately leads to the conclusion that both series converge absolutely.
  • #1
limddavid
6
0

Homework Statement



"Determine whether the following series converge:

[itex]\sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}}[/itex]

and

[itex]\sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}}[/itex]

Homework Equations



The convergence/divergence tests (EXCEPT INTEGRAL TEST):

Ratio
Dyadic
Comparison
P-test
Cauchy Criterion
Root Criterion
Alternating Series Test/Leibniz Criterion
Abel's Criterion

The Attempt at a Solution



My TA said it was helpful to use the Dyadic Criterion to solve series involving logs... I believe this is an exception. It made the equation really convoluted:

[itex]\sum_{n \geq 2} \frac{2^{2k}*k*ln(2)}{(k*ln(2))^{2^{k}}}[/itex]

I'm sure I have to use some combination of the tests, but I kind of need to be pointed in the right direction... I have no idea how to work with that series..

Thank you!
 
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  • #2
limddavid said:

Homework Statement



"Determine whether the following series converge:

[itex]\sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}}[/itex]

and

[itex]\sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}}[/itex]

Homework Equations



The convergence/divergence tests (EXCEPT INTEGRAL TEST):

Ratio
Dyadic
Comparison
P-test
Cauchy Criterion
Root Criterion
Alternating Series Test/Leibniz Criterion
Abel's Criterion

The Attempt at a Solution



My TA said it was helpful to use the Dyadic Criterion to solve series involving logs... I believe this is an exception. It made the equation really convoluted:

[itex]\sum_{n \geq 2} \frac{2^{2k}*k*ln(2)}{(k*ln(2))^{2^{k}}}[/itex]

I'm sure I have to use some combination of the tests, but I kind of need to be pointed in the right direction... I have no idea how to work with that series..

Thank you!

Try the root test; C=lim{n->inf} sup n^(ln(n)/n)/ln(n). Then Let u=ln(n) and substitute this into the root test. Answer should converge to C=0. So the series converges absolutely.
 
  • #3
shaon0 said:
Try the root test; C=lim{n->inf} sup n^(ln(n)/n)/ln(n). Then Let u=ln(n) and substitute this into the root test. Answer should converge to C=0. So the series converges absolutely.

Ok.. I tried to root test, but I'm not sure how I can take the limsup of what I get:

n^(u/n)/u
 
  • #4
limddavid said:
Ok.. I tried to root test, but I'm not sure how I can take the limsup of what I get:

n^(u/n)/u

u=ln(n) → eu2e-u/u and so you get convergence to 0.
 

1. What is the definition of convergence of a series?

The convergence of a series is a mathematical concept that refers to the behavior of a series as its terms are added together. A series is said to converge if the sum of its terms approaches a finite value as the number of terms increases.

2. How can I determine if a series is convergent or divergent?

There are various tests that can be used to determine the convergence or divergence of a series, such as the Comparison Test, the Ratio Test, and the Root Test. These tests involve analyzing the behavior of the terms in the series and determining if they are decreasing or approaching zero.

3. Can a series be both convergent and divergent?

No, a series can only be either convergent or divergent. If the series converges, it means that the sum of its terms approaches a finite value. If the series diverges, it means that the sum of its terms does not approach a finite value and may either tend to infinity or oscillate between different values.

4. What is the significance of convergence of series in real-world applications?

The concept of convergence of series is essential in various fields, such as physics, engineering, and finance. In physics, it is used to determine the behavior of systems and their limits. In engineering, it is used to analyze the stability and performance of systems. In finance, it is used to calculate the present value of future cash flows.

5. Are there any real-world examples of divergent series?

Yes, there are many real-world examples of divergent series. One of the most well-known examples is the harmonic series, which is the sum of the reciprocals of all natural numbers. This series is divergent, meaning that the sum of its terms approaches infinity as the number of terms increases. Other examples include the geometric series with a ratio greater than 1 and the alternating harmonic series.

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