Every Nontrivial Finite MRB Partial Sum Is Irrational

Marvin Ray Burns

Abstract

Let

denote the

th partial sum of the defining series for the MRB constant. We prove that every nontrivial finite partial sum is irrational:

The proof uses the vanishing field trace of each nonrational radical

together with the exact cancellation of the integer parts in odd partial sums. The result applies equivalently to the fractional-part representation

1. Finite MRB partial sums

Let

Since the

term is zero, we have

For every

,

and hence

Thus

We now prove that every partial sum after the trivial first one is irrational.

Theorem 1

For every integer

,

The proof rests on a simple trace property of the individual radicals.

Lemma 1

For every integer

,

Proof

Write the prime factorization of

as

and put

Then

By the definition of

, for every prime

dividing

, the integer

is not an

th power in

. The standard irreducibility criterion for binomials therefore shows that

is irreducible over

. The additional exceptional condition in the binomial criterion when

is automatic here because $c>0$.

Thus

is the minimal polynomial of

. Since $d>1$ and the coefficient of

is zero, the sum of its conjugates is zero. Therefore

Proof of Theorem 1

Fix

and let

and write

For each

, trace transitivity and the preceding lemma give

Now

The elementary alternating sum is

Consequently,

Suppose that

. For a rational number

,

Hence (1) would force

It remains to exclude these two possibilities.

Set

For $x>e$,

Since

the sequence $b_n$ is strictly decreasing for integers

.

First,

because

Thus

.

Next,

We use the rational estimates

They follow respectively from

Therefore

For every odd

,

Every difference in parentheses is positive because $b_n$ is decreasing. Hence

Thus no odd partial sum $S_N$ with

is zero.

For the even indices,

Also,

and for every even

,

because

is odd and

.

Thus every even partial sum is positive, so in particular

Both rational alternatives in (2) are impossible. Therefore

Corollary 1

Every nontrivial finite partial sum in the fractional-part representation

is irrational.

Proof

For every

,

so the fractional-part partial sum is exactly $S_N$. The theorem therefore applies directly.

2. Remark on the infinite limit

The theorem establishes an exact arithmetic property of every finite MRB approximation:

In particular, the MRB constant is approached by a sequence of irrational algebraic numbers, and no nontrivial finite truncation of its defining series is rational.

This fact alone does not prove that the infinite limit

is irrational, since a sequence of irrational algebraic numbers may converge to a rational number.

It does, however, show that rationality can never occur at any finite stage of the defining MRB process.

Optional Wolfram Language verification snippets

Define the partial sums

In[]:=

mrbPartialSum[n_Integer?Positive]:=Sum[(-1)^k(k^(1/k)-1),{k,1,n}]

Equivalent fractional-part form

In[]:=

mrbFractionalPartialSum[n_Integer?Positive]:=Sum[(-1)^kFractionalPart[k^(1/k)],{k,1,n}]

Check equality numerically for small values

In[]:=

Table[N[mrbPartialSum[n]-mrbFractionalPartialSum[n],30],{n,1,12}]

Out[]=

{0,0,0,0,0,0,0,0,0,0,0,0}

Verify positivity and sign pattern for initial cases

In[]:=

Table[{n,N[mrbPartialSum[n],30]},{n,1,12}]

Out[]=

{{1,0},{2,0.414213562373095048801688724210},{3,-0.0280360079343133335199495865704},{4,0.386177554438781715281739137639},{5,0.00644789297756688289167567342327},{6,0.354454047574844550265911987326},{7,0.0339847998187207584565846558261},{8,0.330824354469730424390338773619},{9,0.0543063474604887647059401230358},{10,0.313231759254655975129894229432},{11,0.0696565313422852400501025175221},{12,0.299732036920256493713108013639}}

Trace examples for radicals

Documentation links

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