The MRB constant is a

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

, with decimal expansion . The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999. Burns had initially called the constant "rc" for root constant but, at Simon Plouffe's suggestion, the constant was renamed the 'Marvin Ray Burns's Constant', or "MRB constant". The MRB constant is defined as the upper limit of the

partial sums In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mat ...

s_n = \sum_^n (-1)^k k^

n

grows to infinity, the sums have upper and lower limit points of −0.812140… and 0.187859…, separated by an interval of length 1. The constant can also be explicitly defined by the following infinite sums: :

0.187859\ldots = \sum_^ (-1)^k (k^ - 1) = \sum_^ \left((2k)^ - (2k-1)^\right).

The constant relates to the

divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mu ...

: :

\sum_^ (-1)^k k^.

There is no known

closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...

of the MRB constant, nor is it known whether the MRB constant is algebraic,

transcendental Transcendence, transcendent, or transcendental may refer to: Mathematics * Transcendental number, a number that is not the root of any polynomial with rational coefficients * Algebraic element or transcendental element, an element of a field exten ...

or even

irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...

References

External links

{{Portal, Mathematics
Official site of M.R. Burns, constant's namesake and discoverer
Mathematical constants