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Marvin Ray Burns

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19 years, 254 days
Contact Marvin Ray Burns

 I've been using Maple since 1997 or so.

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These are Posts that have been published by Marvin Ray Burns

MRB Constant V...

Marvin Ray Burns 550 Maple
constant
March 23 2012
0 2

The*MRB*constant = sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity) and sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity) = sum((-1)^n*(n^(1/n)-1), n = 2 .. infinity)

But what can we say about

(∏)(-1)^(n)*(n^(1/(n))-1)?

``

``

Maple does not evaluate it:

evalf(product((-1)^n*(n^(1/n)-1), n = 2 .. infinity))

product: Cannot show that (-1)^n*(n^(1/n)-1) has no zeros on [2,infinity] product((-1)^n*(n^(1/n)-1), n = 2 .. infinity)

 (1)

And perhaps it should not because of the alternating sign;

evalf(product((-1)^n*(n^(1/n)-1), n = 2 .. 10^2))

-0.3908773173e-101

 (2)

evalf(product((-1)^n*(n^(1/n)-1), n = 2 .. 10^3))

-0.7676360791e-1799

 (3)

evalf(product((-1)^n*(n^(1/n)-1), n = 2 .. 10^3+1))

0.5316437097e-1801

 (4)

``

 

Download 3232012.mw

MRB constant U...

Marvin Ray Burns 550 Maple
March 18 2012
0 0

If you use all the convergents of the simple continued fraction of the MRB constant as the terms of a generalized continued fraction, then likewise use the new convergents in another generalized continued fraction, and so on... you arrive at 0.5557531....  For more on this process see https://oeis.org/wiki/Convergents_constant .

MRB constant T...

Marvin Ray Burns 550 Maple
constants
January 13 2012
0 2


Let c=MRB constant -1/2

MRB constant S part 2...

Marvin Ray Burns 550 Maple
constants
January 08 2012
0 0

 

 

 

``

> 

restart; Digits := 64

``

 

````Define s as the following function involving a divergent series.

> 

s := proc (x) options operator, arrow; sum((-1)^n*n^(1/n), n = 1 .. x) end proc

proc (x) options operator, arrow; sum((-1)^n*n^(1/n), n = 1 .. x) end proc

 (1)

``

``

 

 

``The upper limit point of the partial sums, of s is very slowly convergent.

> 

evalf(s(100))

.211329543346941069485035868216520490712148674852018130412747187

 (2)
> 

evalf(s(1000))

.191323989712141370638688981469071803275457219110707245455878532

 (3)
> 

evalf(s(10000))

.188320351076950504638897789942367214051161086517598649780487746

 (4)

 

``

Let mrb be tthe upper limit point of s as x goes to infinity.

``

> 

mrb := evalf(sum((-1)^n*(n^(1/n)-1), n = 1 .. infinity))

.1878596424620671202485179340542732300559030949001387861720046841

 (5)

``

``

``

 

Define f as the following function involving the divergnet series sum((-1)^n*(n^(a/n)-a), n = 1 .. infinity).NULL

``

``

> 

f := proc (a) options operator, arrow; sum((-1)^n*(n^(a/n)-a), n = 1 .. infinity) end proc

proc (a) options operator, arrow; sum((-1)^n*(n^(a/n)-a), n = 1 .. infinity) end proc

 (6)

``

``

 

 

``Let c be the value for a in the neighborhood of 26 such that f(a)=mrb.

c := fsolve(eval(f(x)) = mrb, x = 26)

25.71864739101744668471488151161460875040712539231550975094037406

 (7)

``

``

 

``The average of the upper and lower limit points of the partil sums of f converges much faster than the  upper limit point of the partial sums of s.

> 

evalf((sum((-1)^n*(n^(c/n)-c), n = 1 .. 100)+sum((-1)^n*(n^(c/n)-c), n = 1 .. 101))*(1/2))

.195238896203546569611605945649919224928195587923897718988014700

 (8)
> 

evalf((sum((-1)^n*(n^(c/n)-c), n = 1 .. 1000)+sum((-1)^n*(n^(c/n)-c), n = 1 .. 1001))*(1/2))

.187904922391719396683391551158554482265830937732923110694243700

 (9)
> 

evalf((sum((-1)^n*(n^(c/n)-c), n = 1 .. 10000)+sum((-1)^n*(n^(c/n)-c), n = 1 .. 10001))*(1/2))

.187860182910509428926222275077446745338505139578191116998518780

 (10)

 

Download Jan72012.mw

MRB constant S...

Marvin Ray Burns 550 Maple
constants
January 02 2012
0 15

 


NULL

NULL

Let f(c)= sum((-1)^n*(n^(c/n)-c), n = 1 .. infinity)

NULL

NULL

Then f(1) = the MRB constant:

evalf(eval(sum((-1)^n*(n^(c/n)-c), n = 1 .. infinity), c = 1)) = .1878596425NULL

NULL

NULL

What if we change the value of c and use Levin's u-transform to compute the values for the analytic extension of the sum?

Then can we find values for c such that f(c)=c?

 

evalf(eval(sum((-1)^n*(n^(c/n)-c), n = 1 .. infinity), c = -1.351776595077954)) = -1.351776595 

evalf(eval(sum((-1)^n*(n^(c/n)-c), n = 1 .. infinity), c = 7.020400867228059)) = 7.020400867

evalf(eval(sum((-1)^n*(n^(c/n)-c), n = 1 .. infinity), c = 25.58774196597964)) = 25.58880851

``

As an alalytic extension of the sum is there another value for c such that f(c) = the MRB constant? I haven't found one.

NULL

 

 

``


Download jan022012.mw

 

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