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MRB Constant-C

Marvin Ray Burns's picture
Posted on 2009-02-21 23:19 By Marvin Ray Burns (Maple Rating 2 110)
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Looking at the attached worksheet,

 

View 565_int vs sum for mrb.mw on MapleNet or Download 565_int vs sum for mrb.mw
View file details, it appears that the absolute value, minus 1/2, of the integral of (-1)^x*x^(1/x) from 1 to infinity would equal the partial sum of (-1)^x*x^(1/x) from 1 to where the upper summation is even and growing without bound [0]. Is anyone interested in improving or disproving this conjecture? 

 

 

 

[0] Google "MRB Constant"

 

 

 

 

Comments

Robert Israel's picture

Undefined

Comment on 2009-02-22 01:06 from Robert Israel (Maple Rating 4 2434)

The function f(x) = (-1)^x * x^(1/x) = exp(I*Pi*x + ln(x)/x) is not integrable on [1,infinity): note that f(x) ~ exp(I*Pi*x) as x -> infinity.  What does have a limit as N -> infinity (for integers N) is
int(f(x), x = 1 .. 2*N) .  Similarly, sum(f(n),n=1..2*N) has a limit as N -> infinity.
 

Marvin Ray Burns's picture

Point taken

Comment on 2009-02-23 16:05 from Marvin Ray Burns (Maple Rating 2 110)

 

For x>0and f(x) = (-1)^x * x^(1/x) , the next document, View 565_part 2 sum vs int.mw on MapleNet or Download 565_part 2 sum vs int.mw
View file details , shows the difference of  and the ratio of abs(int(f(x), x = 1 .. 2*N))-1/2  and sum(f(n),n=1..2*N) as integer N -> infinity. The difference seems to go to 0 and the ratio seems to go to 1. However, it would be nice to have a proof of some sort to show that they both converge upon the same number. I have given a simple geometric example of the summing of f in the following: http://marvinrayburns.com/what_is_mrb.mht. Perhaps a similar example of the integrating of f could help make the proof -- that is if it is true.

 

Marvin Ray Burns's picture

not true

Comment on 2009-03-07 18:30 from Marvin Ray Burns (Maple Rating 2 110)

My conjecture is not true. See the following link for some bounds of the value.

http://marvinrayburns.com/latest.html

 

 

Marvin Ray Burns's picture

Some research done

Comment on 2009-12-23 15:14 from Marvin Ray Burns (Maple Rating 2 110)

It appears that RICHARD J. MATHAR has written a paper on this subject of this subject of Int(f(x)) where f(x) = (-1)^x * x^(1/x) = exp(I*Pi*x + ln(x)/x).   arxiv1.library.cornell.edu/PS_cache/arxiv/pdf/0912/0912.3844v1.pdf

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