This is the second half of the main post.
§2
The MRB constant relates to the divergent series:
=
The divergent sequence of its partial sums have two accumulation points with an upper limiting value or limsup known as the MRB constant (CMRB), and a liminf of CMRB-1:
So, out of the many series for CMRB, first analyze the sum prototype or prototypical series, i.e., the conditionally convergent summation, the sum of two divergent ones:
=
Because the Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge, we will quickly derive an absolutely convergent one. (In a sense it is the "completion" of the prototypical series.)
To find its completion, perform the process, add the terms of another conditionally convergent series then subtract their sum.
WHY? and HOW does that produce the absolutely convergent series?
That gives us this absolutely convergent sum:
because
In[167]:= Limit[D[DirichletEta[n], n], n -> 1]
Out[167]= EulerGamma Log[2] - Log[2]^2/2
Next, noting
In[11]:= Series[D[DirichletEta[n], n], {n, Infinity, 1}]
Out[11]=
and perhaps a little more that I don
t understand,
we can even go further and obtain an exponential convergence rate, because of the following identity involving the Dirichlet Eta function derivatives,
In[47]:= Sum[(-1)^n Log[n]/n^x, {n, 1, Infinity}]
Out[47]= 2^-x (2 Log[2] Zeta[x] - 2 Derivative[1][Zeta][x] +
2^x Derivative[1][Zeta][x])
In[48]:= Limit[D[DirichletEta[n], n], n -> x]
Out[48]=
2^(1 - x) Log[2] Zeta[x] + (1 - 2^(1 - x)) Derivative[1][Zeta][x]
In[46]:= FullSimplify[
Sum[(-1)^n Log[n]/n^x, {n, 1, Infinity}] ==
Limit[D[DirichletEta[n], n], n -> x]]
Out[46]= True
The completed absolutely convergent and rapidly convergent sum:
GEMINI seems to recall it by the following draft of as summary:
The passage discusses the MRB constant (CMRB) and its connection to a
specific series representation. Here's a breakdown of the key points:
(1.) MRB Constant and Divergent Series:
The MRB constant relates to a divergent series, meaning the sum of its
terms doesn't approach a specific value as the number of terms tends
to infinity. However, the partial sums of this series (sums up to a
certain number of terms) have two interesting accumulation points: An
upper limiting value (limsup) called the MRB constant (CMRB). A lower
limiting value (liminf) one less than CMRB.
(2.) Analyzing the Prototypical Series:
The passage focuses on a specific series representation for CMRB,
which is the sum of two other divergent series. The Riemann series
theorem states that a conditionally convergent series (where the signs
of terms alternate) can be rearranged to converge to any desired value
or diverge altogether.
(3.) Deriving an Absolutely Convergent Series:
To overcome the limitations of the original series, the passage
proposes a method to find its "completion." This involves adding and
subtracting terms from conditionally convergent series in a specific
way. The goal is to obtain an absolutely convergent series, where the
sum of the absolute values of the terms converges to a finite value.
( 4.) The Absolutely Convergent Series:
By manipulating the original series and utilizing properties of the
Dirichlet Eta function (a special function in number theory), the
passage arrives at an absolutely convergent series representation for
CMRB. This new series not only converges but also exhibits an
exponential convergence rate, meaning the sum approaches its limit
much faster as the number of terms increases.
(5.) Utilizing the Dirichlet Eta Function:
The passage leverages properties of the derivative of the Dirichlet
Eta function to establish the equivalence between the original series
and the limit of that derivative. This equivalence helps demonstrate
the validity of the derived absolutely convergent series. In essence,
the passage presents a technique to transform a problematic series
representation of the MRB constant into a well-behaved, absolutely
convergent series with a faster convergence rate. This new series
representation provides a more reliable and efficient way to calculate
the MRB constant.
§9 for a full proof.
Here is how the speedily convergent sum (in midnight blue (Look closely!)) compares with the prototype (in yellow) to the precise value of the MRB constant (CMRB) (in red).
Concerning the sum prototype for CMRB
§3. Is the series convergent?
Proof of its convergence by Leibniz's criterion is shown next.
LEMAS:
PROOF
Below, it is shown by the Squeeze theorem (sandwich theorem) and by plotting the following series, the qualifications for the Leibniz criterion (alternating series test) are satisfied for the MRB constant (CMRB) in
CMRB
by showing a(n)=(n^{1/n}-1)>0 is monotonically decreasing for
$n≥3$ and have a limit as n goes to infinity of zero. Of course, , converges, and the sum of two convergent series converges.
To use
- Notice that we saw, for n greater than 1, the derivative equals zero
solely at the value of e, and there are no additional critical
points at which the plot ceases to decline. Thus, the function,
n^(1/n)-1 is monotonic decreasing as n goes to infinity. .
- We also saw, the limit of n^(1/n)-1 is zero:
To confirm the above limit, here is a direct proof from
that
because
By utilizing the Squeeze theorem (also known as the Sandwich theorem) and plotting techniques, it have been demonstrated that the Leibniz criterion (alternating series test) is valid for the alternating sum of a(n) = (n^(1/n) - 1) > 0. Moreover, for n greater than 1, the derivative equals zero solely at the value of e, and there are no additional critical points at which the plot ceases to decline. As a result, a(n) = (n^(1/n) - 1) > 0 decreases monotonically for n greater than or equal to 3 and have a limit of zero as n approaches infinity. Ultimately, the series converges, and the sum of two convergent series is also convergent. Thus, the CMRB Sum prototype series is convergent. ∎
§4
Next, ask and observe,
As shown soon ( "As for efficiency" ).
§5. Is that series, absolutely convergent?
The following criterion works remarkably well in determining its absolute convergence.
Plot[{n^(1/n) - 1, 1/n}, {n, 1, Infinity},
PlotLegends -> LineLegend["Expressions"]]
...showing its terms are larger than those of the divergent Harmonic Series.
§6
As for efficiency, this discussion presents several series that converge much faster for CMRB. Here are a few of their convergence rates. The following expressions show the sum followed by the closeness to zero of their result after a certain number of partial summations.
ALG 1 proposed here
For how more efficient forms compare,
"the rate of convergence" of 3 major forms.
§7.
This discussion is not crass bragging; it is an attempt by this amateur to share my discoveries with the greatest audience possible.
Amateurs have made a few significant discoveries, as discussed in here.
This amateur have tried my best to prove my discoveries and have often asked for help. Great thanks to all of those who offered a hand!
As I went more and more public with my discoveries, making several attempts to see what portions were original, I concluded from these investigations that the only original thought was the obstinacy to think anything meaningful could be found in the infinite sum shown next.
Nonetheless, someone might have a claim to this thought to whom it have not given proper credit. Hence (apologies!) The last thing needed is another calculus war, this time for a constant. However, as Newton said concerning Leibniz's claim to calculus, anyone's thought was published after his, “To take away the Right of the first inventor and divide it between me and that other would be an Act of Injustice.” [Sir Isaac Newton, The Correspondence of Isaac Newton, 7 v., edited by H. W. Turnbull, J. F. Scott, A. Rupert Hall, and Laura Tilling, Cambridge University Press, 1959–1977: VI, p. 455]
Here is what Google says about the MRB constant as of August 8, 2022, at
https://www.google.com/search?q=who+discovered+the+%22MRB+constant%22
(the calculus war for CMRB)
CREDIT
https://soundcloud.com/cmrb/homer-simpson-vs-peter-griffin-cmrb
'
From Wikipedia, the free encyclopedia:
'
The calculus controversy (German: Prioritätsstreit, "priority dispute") was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had invented calculus.
(Newton's notation as published in PRINCIPIS MATHEMATICA [PRINCIPLES OF MATHEMATICS])
(Leibniz's notation as published in the scholarly journal Acta Eruditorum [Reports of Scholars])
Whether or not we divide the credit between the two pioneers,
said one thing that distinguishes their finds from the work of their antecedents:
Newton came to calculus as part of my investigations in physics and
geometry. I viewed calculus as the scientific description of the
generation of motion and magnitudes. In comparison, Leibniz focused on
the tangent problem and came to believe that calculus was a
metaphysical explanation of the change. Importantly, the core of their
insight was the formalization of the inverse properties between the
integral and the differential of a function. This insight had been
anticipated by their predecessors, but they were the first to conceive
calculus as a system in which new rhetoric and descriptive terms were
created.[24] Their unique discoveries lay not only in their
imagination but also in their ability to synthesize the insights
around them into a universal algorithmic process, thereby forming a
new mathematical system.
Like as Newton and Leibniz created a new system from the elaborate, confusing structure designed and built by their predecessors, Marvin Ray Burns' forerunners studied series for centuries leading to a labyrinth of sums, and then, came a "new scheme" for the CMRB "realities" to escape and stand out as famous in their own right!
§8.
is defined in the following 31 places, most of which attribute it to my curiosity.
ค่าคงที่ลุ่มแม่น้ำโขง (in Thai);
ar.wikipedia.org/wiki/ (In Arabic);
https://calculla.com 300+ calcullators online (in Polish)
Constante MRB (in French);
Constanta MRB - MRB constant (in Romanian);
http://constant.one/ ;
Crandall, R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012,ISBN-10 : 193563819X ISBN-13: 978-1935638193;
Crandall, R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012;
https://en-academic.com/, Wikipedia, Mathematical constant;
Encyclopedia of Mathematics (Series #94);
Engineering Tools of the Iran Civil Center (translated from Persian), an international community dedicated to the construction industry, ISSN: 1735–2614;
Etymologie CA Kanada Zahlen" (in German). etymologie.info;
Finch, S. R. Mathematical Constants, Cambridge, England:
Cambridge University Press, p. 450, 2003, ISBN-13: 978-0521818056, ISBN-10: 0521818052;
Finch's original essay on Iterated Exponential Constants;
Finch, Steven & Wimp, Jet. (2004). Mathematical constants. The Mathematical Intelligencer. 26. 70-74. 10.1007/BF02985660;
Journal of Mathematics Research; Vol. 11, No. 6; December 2019 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education;
Library of General Functions (LGF) for SIMATIC S7-1200
Mauro Fiorentina’s math notes (in Italian);
https://mathparser.org/mxparser-math-collection/constants/
MATHAR, RICHARD J. "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^(1/x) BETWEEN 1 AND INFINITY" (PDF). arxiv. Cornell University;
Mathematical Constants and Sequences a selection compiled by
Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan’s Library,
ISSN 2421-1230, Vol.II;
"Matematıksel Sabıtler" (in Turkish). Türk Biyofizik Derneği;
MathWorld Encyclopedia;
MRB常数 (in Chinese);
mrb constantとは 意味・読み方・使い方 ( in Japanese);
MRB константа (in Bulgarian);
OEIS Encyclopedia (The MRB constant);
Patuloy ang MRB - MRB constant (in Filipino)
Plouffe's Inverter;
the LACM Inverse Symbolic Calculator;
The On-Line Encyclopedia of Integer Sequences® (OEIS®) as
A037077, Notices Am. Math. Soc. 50 (2003), no. 8, 912–915, MR 1992789 (2004f:11151);
Wikipedia Encyclopedia (in English):
Here are the MRB constant's Wikipedia views vs those of the famous Euler's constant, e:
§9.
= B =
and from Richard Crandall ( "** From Bing AI:") in 2012 courtesy of Apple Computer's advanced computational group, the following computational scheme using equivalent sums of the zeta variant, Dirichlet eta:
The expressions and denote the mth derivative of the Dirichlet eta function of m and 0, respectively.
The c_{j}'s are found by the code,
N[ Table[Sum[(-1)^j Binomial[k, j] j^(k - j), {j, 1, k}], {k, 1, 10}]]
(* {-1., -1., 2., 9., 4., -95., -414., 49., 10088., 55521.}*)
** From Bing AI:
--
Crandall's first "B" is proven below by Gottfried Helms, and it is proven more rigorously, considering the conditionally convergent sum, afterward. Then the formula (44) is a Taylor expansion of eta(s) around s = 0.
At here, we have the following explanation.
The integral forms for CMRB and MKB differ by only a trigonometric multiplicand to that of its analog.
In[182]:= CMRB =
Re[NIntegrate[(I*E^(r*t))/Sin[Pi*t] /. r -> Log[t^(1/t) - 1]/t, {t,
1, I*Infinity}, WorkingPrecision -> 30]]
Out[182]= 0.187859642462067120248517934054
In[203]:= CMRB -
N[NSum[(E^( r*t))/Cos[Pi*t] /. r -> Log[t^(1/t) - 1]/t, {t, 1,
Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 37],
30]
Out[203]= 5.*10^-30
In[223]:= CMRB -
Quiet[N[NSum[
E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -> Log[t^(1/t) - 1]/t, {t,
1, Infinity}, Method -> "AlternatingSigns",
WorkingPrecision -> 37], 30]]
Out[223]= 5.*10^-30
In[204]:= Quiet[
MKB = NIntegrate[
E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -> Log[t^(1/t) - 1]/t, {t,
1, I*Infinity}, WorkingPrecision -> 30, Method -> "Trapezoidal"]]
Out[204]= 0.0707760393115292541357595979381 -
0.0473806170703505012595927346527 I
\
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