it follows that
For what value of x is f(x)=x^(1/x) a maximum?


x=e and gives a maximum of e^(1/e)=1.444667861...

High precision evaluation of the series $\sum_{n=3}^\infty
(-1)^n (1-n^{1/n})$, URL (version: 2019-07-16):
where A is the Glaisher-Kinkelin constant.
formulas (11)+(16)+(19).
Dark Malthorp
What are some working models that are a fit the formula for the
MRB constant?, URL (version: 2020-01-12):
where  η is the Dirichlet eta function. Both should be valid for all z∈C.

Google Scholar
MRB constant  (CMRB)
5,555,555 digit computation of CMRB
(MRB supercomputer output at 18:50:08 am EDT  |  Friday, July 3, 2020)
Start time is Fri 19 Jun 2020 22:20:08.
Iterations (iter.) required: 7369998
Will give 3599 time estimates, each more accurate than the previous.
Will stop at 7370752 iterations to ensure precision of around 5583332 decimal
1132544 iter. done in 1.161*10^6 s. Ave. 0.97566iter./s. Should take 87.429 days
or 7.554*10^6s. Finish Tue 15 Sep 2020 08:37:29.

{15.3654,% done.}

MRB supercomputer
Processors: Intel Core i9-9900K (5.0 GHz Turbo) (16-Thread) (8-Core) 3.6
GHz. CPU Boost: Stage 2: Overclock CPU - Up to 5.1GHz on All CPU Cores at
3200 Mhz RAM Extreme Cooling: H20: Stage 2: Corsair H115i PRO - 280mm
Liquid CPU Cooler  with the following two
exterior nodes: 4.7 GHz on all Intel
6 cores at 3000 MHz RAM and 4 cores of 3.6 GHz at 2400 MHz RAM
<- by Fubini’s theorem for double series along with->
<- by a Taylor expansion of η(s) around s = 0
<- as pointed out by J. Borwein,         is the j-th Taylor
coefficient in the expansion of             about x = 0.
Try to break these CMRB Computational
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Try to break these CMRB Computational Records(Click for the full amazing ride!)
This conditionally convergent series is transformed
into the following
absolutely convergent two by
Richard Crandall (2012).
These limits are transformed into the
infinite series.
Collecting the terms of this series gives us
the following
conditionally convergent one.
(η is the Dirichlet eta function.)
Finch, Steven R. (2003). Mathematical
Constants. Cambridge, England: Cambridge
University Press. p. 450. ISBN 0-521-81805-2
First published in book form by
Gottfried Helms (, MRB constant
proofs wanted, URL (version: 2015-12-24):
Given that C MRB =

The MRB constant is defined at After a lot of looking I found a
connection between the MRB constant and applied math:  

The MRB constant is ∑(−1)^k(k^(1/k)−1), and that k^(1/k)−1 is the interest rate to multiply an investment k times in
k periods -- as well as other growth models involving the more general expression (1+k)^n --
since ((k^(1/k)−1)+1)^k|k∈Z+=k. and ((k^(1/n)−1)+1)^n|n∈Z+=k.

We can say, the result of summing, with alternating signs, the interest rate to multiply an investment k times in k
periods (or the equivalent growth model) could be the end "growth" rate resulting from growth, following decay,
following growth, ad infinitum.  

Since         = 3, consider
Notice this real life application.