The Geometry of the MRB constant

The MRB constant is the upper limit point of the sequence of partial sums defined by:

\displaystyle S(x) = \sum_{n=1}^{x}{(-1)^n n^\frac{1}{n}}.

The goal of this article is to show that the MRB constant is geometrically quantifiable. To “measure” the MRB constant, we will consider a set, sequence and alternating series of the nth roots of n. Then we will compare the length of the edges of a special set of hypercubes or n-cubes which have a content of n. (The two words hypercubes and n-cubes will be used synonymously.)

Finally we will look at the value of the MRB constant as a representation of that comparison, of the length of the edges of a special set of hypercubes, in units of dimension 1/ (units of dimension 2 times units of dimension 3 times units of dimension 4 times etc.). For an arbitrary example we will use units of length/ (time*mass* density*…).

A Countable Infinite Set

Consider, r, the set of roots of positive integers of the form r = n^(1/n). Of course the elements of this set are of the form x^(1/y). However what is not obvious is the geometric interpretation of x^(1/y). At least as far as natural number valued (x and y)>0 are concerned, x represents the content of an n-cube and y represents its dimension. That is a geometric interpretation of x^(1/y) as far as natural number valued (x and y)>0 are concerned. For instance we take a cube of any given volume and find the length of one of its sides. Let’s suppose the volume was 8 units^3. What would be the length of one of its sides? We might easily deduce that the length is 2. To confirm this answer we simply construct a cube of 2 linear units in length as in Diagram 1 and find its volume.

Diagram 1

The volume in units3 of the cube in Diagram 1 is indeed 2*2*2 = 8.
Now we look at the previous sentence with x^(1/y) in mind. 8^(1/3) = 2 implies the volume of the cube in Diagram 1 raised to the power of the reciprocal of its dimension equals the length of one of its sides. That is a geometric interpretation of x^(1/y) as far as natural number valued x and y > 0 are concerned.

Open Question

What is the geometric interpretation of x^(1/y) for all real values of x,y > 0?

Sequence and Alternating Series

Now we will consider the sequence of roots of positive integers of the form r = {n^(1/n)} = {1^(1/1), 2^(1/2), 3^(1/3), …}. Then we will add the elements of r in the alternating series

\displaystyle L = \sum_{n=1}^{\infty}{(-1)^n r(n)} = \sum_{n=1}^{\infty}{(-1)^n n^\frac{1}{n}}.

Concerning the partial sums of L, we remember

\displaystyle S(x)= \sum_{n=1}^{x}{(-1)^n n^\frac{1}{n}}

and we find, S(x) is divergent as x goes to infinity. However S(2x) and S(2x+1) are both convergent as x goes to infinity and the difference between S(2x) and S(2x+1) also converges.

Special Hypercubes

In geometric terms: As in Diagram 2, we give each n-cube a content equal to its dimension1,

Diagram 2

so that we have a line segment of 1 linear unit, a square of 2 square units, and so forth.

The last cube in the diagram may be just an invention of the imagination because who has ever heard of a hypercube of unbounded dimension with unbounded content? Furthermore, there seems to be a paradox invoked when taking n-cubes as n->∞. While n is still a number the content of the n-cube or hypercube is defined in the specific unit of choice, whether the unit is inches, meters or what not and the resulting length of an individual edge is also defined in the same unit and is computed as shown above. “The [content] of the [n-]cube raised to the power of the reciprocal of its dimension equals the length of one of its [edges].”

However, when we arrived at the hypercube of unbounded dimension where n is no longer a number, the assigned “unbounded content” could be meant to be in units of feet, let’s say; while the resulting length of an individual edge is \displaystyle \lim_{u \to \infty}{u^\frac{1}{u}} = 1 which could be in feet or any other unit of length because there are the same amount of inches or meters or any other unit of length in infinity feet as there are feet. To avoid the resulting ambiguity we will look at the sequence of roots of positive integers of the form r = {n^(1/n)} = {1^(1/1), 2^(1/2), 3^(1/3), …} as being analogous to the clopen interval [1, ∞).

A Sum of the Series

Above it is mentioned, “Add the elements of r in the alternating series

\displaystyle L = \sum_{n=1}^{\infty}{(-1)^n r(n)} = \sum_{n=1}^{\infty}{(-1)^n n^\frac{1}{n}}.

To show that geometrically we do the following: as in Diagram 3, on the y,z-plane line up an edge of each n-cube or hypercube. The numeric values displayed in the diagram are the partial sums of S(x) = S(2u) where u is a positive integer:

\displaystyle S(x)=\sum_{n=1}^{x}{(-1)^n n^\frac{1}{n}}.

Diagram 3

Diagram 3

Notice a directed line segment is moved from the origin down the (z or y=0)-axis. Then at the y=1δ axis another one is moved up 2^(1/2) units. Then at the y=2δ axis yet another one is moved down 3^(1/3) units, etc. It does not matter whether δ is one or any other real value; there still are an infinite number of y-valued axes with matching directed line segments.

This is hard to understand; but we may say metaphorically that Diagram 3 is the path along the units of a particle moved 1 inch down in 2^(1/2) seconds, losing 3^(1/3) units of mass with density that increases 4^(1/4) units etc. The resulting position and condition of the particle is represented by

\displaystyle M = \lim_{u \to \infty}{\left ( \sum_{n=1}^{2u}{(-1)^n n^\frac{1}{n}} \right )}.

The MRB Constant

As the dimension and the content of a hypercube, both go to infinity we have the following: First, in Diagram 2 the difference between the length of an edge of the hypercube with content 2n+1, and an edge of the hypercube with content 2n, goes to the constant value,

\displaystyle \lim_{n \to \infty}{\left ( (2n +1)^\frac{1}{2n+1}-(2n)^\frac{1}{2n} \right )} = 0.

So as n goes to infinity, the length of an edge of the hypercube with content 2n and an edge of the hypercube with content 2n+1 become closer to being the same. Second, in Diagram 3 an edge of each n-cube is arranged on y-valued axes in such a way that

\displaystyle \sum_{n=0}^{\infty}{\left ( (2n +1)^\frac{1}{2n+1}-(2n)^\frac{1}{2n} \right )} = \lim_{u \to \infty}{\left ( \sum_{n=1}^{2u}{(-1)^n n^\frac{1}{n}} \right )}.

M is the MRB constant2.

A numerical approximation of M can be computed by the following summation

\displaystyle \sum_{n=1}^{\infty}{(-1)^n \left (n^\frac{1}{n}-1 \right )},

which sum converges3 (see Diagram 4), while

\displaystyle \sum_{n=1}^{\infty}{(-1)^n n^\frac{1}{n}}

diverges4, as mentioned above.

Diagram 4

Diagram 4

One should use acceleration methods when computing a numerical approximation of the MRB constant because it can be shown that one must sum a number in the order of 10^(n+1) iterations of (-1)^n*(n^(1/n)-1) to get n accurate digits of the MRB Constant. However, using a convergence acceleration of alternating series algorithm of Cohen-Villegas-Zagier one can compute the first 60 digits in only 100 iterations5.

In Diagram 4a both the lim sup6 and the lim inf converge upon the MRB constant, while in 4b only the lim sup converges upon it with the lim inf converging upon MRB constant-1. The MRB constant is Sloane’s On-Line Encyclopedia of Integer Sequences id:A0370777. More information, including a brief but documented history, can be found in Wikipedia8.

Summary

In retrospect, the geometry used here, particularly in diagram3, is transdimensional and thus we find it hard to understand through the previous experiences of our senses. (To examine its geometry we used edges from hypercubes of many dimensions.) However, considering the various temporal-spatial dimensions that affect our universe as proposed in some theories9 is there some significance to the MRB constant in our daily lives? Nevertheless, we have seen that the value of the MRB constant is geometrically quantifiable; it is the lim sup of the sequence that represents a particle traveling along a directed line segment that is moved 1 unit from the origin down the z-axis; at the y=1δ moved up 2^(1/2) units’ at the y=2δ axis moved down 3^(1/3) units etc. Whether the units are theoretical (as in units of length/ (time*mass* density*…)) or proposed temporal-spatial dimensions, the resulting z value of the particle’s position and condition

\displaystyle M = \lim_{u \to \infty}{\left ( \sum_{n=1}^{2u}{(-1)^n n^\frac{1}{n}} \right )}.

This article is released under the Creative Commons Attribution-Share Alike 3.0 Unported license.

About the author

Marvin Ray Burns has indulged in math research as a hobby since 1994. Having had only one college course, most of his discoveries were simply learning the basics of math. He has cataloged many of his early investigations at http://math2.org/mmb/search?query=Marvin. One of his ideas has served some purpose in the math world; that is the MRB constant. Since the discovery of the MRB constant, at least one major mathematics software company has found that it was useful to fix problems found while computing the MRB constant. Another major company changed the functionality of its sum function in such a way as to be able to compute the digits of the MRB constant shortly after its discovery. Mr. Burns has submitted a few integer sequences based on his explorations of the MRB constant; see http://www.research.att.com/~njas/sequences/?q=A037077. A siding applicator by profession, he presently takes various undergraduate courses at IUPUI in the hopes of obtaining a degree in pure math.

References

[1] http://www23.wolframalpha.com/input/?i=n-cube
[2] S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 450.
[3] http://mathworld.wolfram.com/MRBConstant.html
[4] http://mathworld.wolfram.com/notebooks/Constants/MRBConstant.nb
[5] http://arxiv.org/abs/0912.3844
[6] http://en.wikipedia.org/wiki/Upper_limit
[7] http://oeis.org/A037077
[8] http://en.wikipedia.org/wiki/MRB_constant
[9] http://arxiv.org/abs/hep-ph/9803466

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8 Comments

  1. John F. McGowan November 23, 2010
    • Marvin November 24, 2010
    • Stephane Rose April 11, 2013
      • Stephane Rose April 11, 2013
  2. John F. McGowan November 24, 2010
  3. Marvin November 24, 2010
  4. John F. McGowan November 25, 2010
  5. sanayaaaaaaaaaaaa December 6, 2014

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