Series Expansion of $\frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}$

In summary, the conversation discusses the asymptotic behavior of a given formula, and the use of a series expansion to simplify the denominator for easier analysis. The speaker also mentions using WolframAlpha to perform the expansion, but notes that the result still contains a logarithm term. They request assistance in understanding how the result was produced.
  • #1
ELW
1
0
Hi,

I'm studying the asymptotic behavior (n -> infinity) of the following formula, where k is a given constant.
[tex]\frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}[/tex]

I'm trying to do a series expansion on this equation to give the denominator a simpler form so that it is easier to make an asymptotic analysis.

I used mathematica/wolframalpha to expand the formula (the documents say Taylor expansion is used).
http://www.wolframalpha.com/input/?i=1/(n^(k+(k+++1)/(2+n))+(2+k+n+-+k+(1+++k)+Log[n])^2)

However in series expansion at n -> infinity, the result still has log n. This is actually a form I prefer, compared to the form [tex] a_0 + a_1 x + a_2 x^2 + ...[/tex] Does anyone see how the result is produced? Any help is much appreciated. Thanks.
 
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  • #2
ELW said:
Does anyone see how the result is produced?

I suggest that you post the result that your are talking about.
 

What is the equation for the series expansion of $\frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}$?

The series expansion of this equation is given by: $\sum_{n=1}^\infty \frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}$

What is the significance of the index $k$ in the equation?

The index $k$ represents the power to which the variable $n$ is raised in the denominator of the fraction. It affects the rate at which the terms in the series converge or diverge.

What are the applications of this series expansion in science?

This series expansion is commonly used in mathematics and physics to approximate the behavior of certain functions. It is also used in various fields such as statistics, engineering, and economics to solve complex problems.

How is this series expansion derived?

The series expansion is derived using the Taylor series method, which involves representing a function as an infinite sum of its derivatives evaluated at a specific point.

What are the limitations of this series expansion?

The series expansion may not always converge for certain values of $k$ and $n$, leading to inaccurate results. It is also limited to functions that can be represented by a Taylor series.

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