User:Marburns/sandbox

Marvin R. Burns, the constant's original investigator, in 1999

This mathematical constant is sometimes called the MRB constant or MRB.. MRB stands for Marvin Ray Burns. Being a sum of irrational numbers its irrationally remains an open problem.

The numerical value of the constant, truncated to 6 decimal places, is

0.187859… (sequence A037077 in the OEIS).

Definition

First 100 partial sums of (-1)^k (k^{1/k} - 1)

The constant is related to the following divergent series:

$\sum _{k=1}^{\infty }(-1)^{k}k^{1/k}.$

Its partial sums

$s_{n}=\sum _{k=1}^{n}(-1)^{k}k^{1/k}$

are bounded so that their limit points form an interval [−0.812140…,0.187859…] of length 1.. The upper limit point 0.187859… is what is sometimes called the MRB constant. The constant can be explicitly defined by the following infinite sums:

$0.187859\ldots =\sum _{k=1}^{\infty }(-1)^{k}(k^{1/k}-1)=\sum _{k=1}^{\infty }\left((2k)^{1/(2k)}-(2k-1)^{1/(2k-1)}\right).$

There is no known closed-form expression of this constant.

History

Marvin Ray Burns published his discovery of the constant in 1999. The discovery is a result of a "math binge" that started in the spring of 1994. Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant. At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999. Since then it has been added to tables of constants in a few countries, including Turkey, Iran, Germany. and the United States.

External links

• (sequence A037077 in the OEIS)

Category:Mathematical constants Category:Number theory

Marvin R. Burns, the constant's author, in 1999

The MRB constant, named after Marvin Ray Burns, is a mathematical constant for which no closed-form expression is known. It is not known whether the MRB constant is algebraic, transcendental, or even irrational.

The numerical value of MRB constant, truncated to 6 decimal places, is

0.187859… (sequence A037077 in the OEIS).

Definition

MRB First 100 points

The MRB constant is related to the following divergent series:

$\sum _{k=1}^{\infty }(-1)^{k}k^{1/k}.$

Its partial sums

$s_{n}=\sum _{k=1}^{n}(-1)^{k}k^{1/k}$

are bounded so that their limit points form an interval [−0.812140…,0.187859…] of length 1. The upper limit point 0.187859… is what is known as the MRB constant.

The MRB constant can be explicitly defined by the following infinite sums:

$0.187859\ldots =\sum _{k=1}^{\infty }(-1)^{k}(k^{1/k}-1)=\sum _{k=1}^{\infty }\left((2k)^{1/(2k)}-(2k-1)^{1/(2k-1)}\right).$

There is no known closed-form expression of the MRB constant.

History

History Marvin Ray Burns published his discovery of the constant in 1999. The discovery is a result of a "math binge" that started in the spring of 1994. Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant. At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999. Since then it has been added to tables and lists of constants in a few countries, including Turkey, Iran, Germany. , the United States. and Italy

External links

• Official site of M.R. Burns, constant's author

Iterated exponentials are an example of an iterated function system based on $x^{x}$. Such systems have induced some interesting mathematical constants and interesting fractal properties based on its generalization to the complex plane.

Inverse

In fact, $f(x)=x^{x}$ does have an inverse

$x=f^{-1}(y)=y^{({\frac {1}{y}})^{({\frac {1}{y}})^{\dots }}}$

which is well-defined for $y\in [({\frac {1}{e}})^{\frac {1}{e}},e^{e}]$

This has induced interest in the function $x^{\frac {1}{x}}$, which has similar limiting properties to $x^{x}$.

Convergence

By an old result of Euler, repeated exponentiation convergence for real values inbetween $e^{-e}$ and $e^{\frac {1}{e}}$.

Calculation of Iterated Exponential

In certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.

Connection to Lambert's Function

If one defines

$h(z)=z^{z^{z^{\dots }}}$

for such $z$ where such a process converges,

Then $h(z)$ actually has a closed form expression in terms of a function known as Lambert's function which is defined implicitly via the following equation:

$W(z)e^{W(z)}=z$

Namely, that

$h(z)={\frac {W(-log(z))}{-log(z)}}$

This can be seen by inputting this definition of $h(z)$ into the other equation that $h(z)$ satisfies, $h(z)=z^{h(z)}$.

Iteration on the Complex Plane

The function may also be extended to the complex plane, where such a map tends to display interesting fractal properties.

Of particular interest is evaluation of the constant

$i^{i^{i^{\dots }}}$

Which does indeed converge and has been evaluated as

$~=0.43828+0.36059i$

<ref>Galidakis, I. N. (2004). On an application of Lambert's W function to infinite exponentials. Complex Variables, Theory and Application: An International Journal, 49(11), 759-780.</math>

Category:Mathematical constants Category:Number theory