Advanced Search
All Posts : MaplePrimes Commons General Technical Discussions

MRB constant S part 2

Marvin Ray Burns 550 Maple
January 08 2012
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

 

 

 

marvinrayburns.com
constants

Please Wait...