User:Marburns/sandbox



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….

Definition
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.

Definition
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

Minimum and Injectivity
We note that $$f(x)=x^x$$ is concave up with minimum

$$(x,f(x))=(\frac{1}{e},(\frac{1}{e})^{\frac{1}{e}})$$

or

=(0.3678794412...,0.6922006276...)

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

$$~=0.43828+0.36059i$$

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.

Category:Mathematical constants Category:Number theory