Consider m, the MRB Constant. 0.187859 <m<0.187860.
One must sum a number in the order of 10^(n+1) iterations of (-1)^n*(n^(1/n)-1) using exact
arithmetic to get n accurate digits of the MRB Constant. See table of convergence[0]. For
example, it takes 223 iterations to get one accurate digit, 37,682 iterations to get three accurate
digits, 149,473 iterations to get four accurate digits, more than 3,000,000 iterations to get five
accurate digits and slightly more than 20,000,000 iterations to get 6 accurate digits. As of March
16, 2008 the world record was 200,000 accurate digits. This represents 10^200,001iterations. At
1 trillion iterations per second it would take 3.168808781402895…* 10^199,881 google years.
However it took only 0.0964612 years. This was made possible because of an algorithm found in
the paper called Convergence acceleration of alternating series.[1] The quickness at which the
200,000 digits was computed represents 3.285060937880398… *10^ 199,982 iterations per
second.
[0]
Table of Convergence
After 10^1 iterations: 0.3
After 10^2 iterations: 0.21
After 10^3 iterations: 0.191
After 10^4 iterations: 0.1883
After 10^5 iterations: 0.18792
After 10^6 iterations: 0.187867
After 10^7 iterations: 0.1878604
After 10^8 iterations: 0.18785973
After 10^9 iterations: 0.187859653
After 10^10 iterations: 0.1878596436
After 10^11 iterations: 0.18785964259
After 10^12 iterations: 0.187859642476
After 10^13 iterations: 0.1878596424636
After 10^14 iterations: 0.18785964246223
After 10^15 iterations: 0.187859642462084
After 10^16 iterations: 0.1878596424620690
After 10^17 iterations: 0.18785964246206732
After 10^18 iterations: 0.187859642462067141
After 10^19 iterations: 0.1878596424620671224
After 10^20 iterations: 0.18785964246206712048
After 10^21 iterations: 0.187859642462067120273
After 10^22 iterations: 0.1878596424620671202511
After 10^23 iterations: 0.18785964246206712024878
After 10^24 iterations: 0.187859642462067120248546
After 10^25 iterations: 0.1878596424620671202485208
After 10^26 iterations: 0.18785964246206712024851823
After 10^27 iterations: 0.187859642462067120248517965
After 10^28 iterations: 0.1878596424620671202485179373
After 10^29 iterations: 0.18785964246206712024851793439
After 10^30 iterations: 0.187859642462067120248517934089
[1] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.
em/1046889587
Convergence of the MRB Constant
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