I must have missed something in this post. Perhaps you could help me understand what some of the insights are?

You started off with something like,

F1:=sum((-1)^n*(n^(1/n)-1),n=1..x):
F2:=sum((-1)^n*(n^(1/n)-2),n=1..x):
F3:=sum((-1)^n*(n^(1/n)-3),n=1..x):

And then you noticed that for each second set of values for these, ie.for x=2,4,6, the F1, F2, and F3 would agree.

But that's just because when x is any even positive integer all three of F1, F2, and F3 simplify immediately to sum((-1)^n*n^(1/n),n = 1 .. infinity);

Now, Maple can show that directly, with,
simplify([F1,F2,F3]) assuming x::even;

You wrote, "Notice that the blue, red and green graphs meet at least one y-value. As x goes to infinity, that y-value is the value of the MRB Constant." But isn't that just the same as the limits of any subsequences of the function coloured green itself, as x goes to infinity? What do the red and blue curves add, for illustrating the behaviour of the green curve?

So it seems that you've named sum((-1)^n*n^(1/n),n=1..infinity) . Is that right?

What do you think of the value that Maple gives when one takes evalf() of that expression? Do you think that the green curve has some subsequences that converge to both a positive number and a negative number? If so, how do you think that those relate to the evalf() result?

I couldn't follow what the summations mean when the index ranges from 1 to sqrt(3). I had thought that the dummy index n was representing positive integers. What does this mean,
sum((-1)^n*n^(1/n),n = 1 .. 27*6^(1/2))));

Thanks very much,
acer

I show in the worksheet, mrbgraphs a, that the constant of concern (the MRB Constant) is indeed a limiting point at Infinity for one member of the family. Since all members have equal partial sums at even intervals, they all converge to the mrb in the following sense.

Last update, Firday Dec 17, 7:17PM

restart;

f1:=x->sum((-1)^n*(n^(1/n)-1),n=1..x);

Technically, as x goes to infinity, f1(x) does converge. n^(1/n)->0 as n->infinity. Therefore, the function that describes the local maximums of f:

aslo converges.

Because the function that describes the local maximums of f converges and because the limit(x^1/x,x=infinity)=1, the function that describes the local minimums of f:

also converges.

limit(x^1/x,x=ininity);

As for the plots I was concerned about in mrbgraphs 1, I can focus solely upon f1 since we know it converges. I will look at the real and imaginary parts and the absolute value of f1.

with(plots):

Warning, the name changecoords has been redefined

Inf:=20:
P0(Inf):=plot(Re(f1(x)),x=1..Inf,color=green,tickmarks=[21,3],style=point):
P1(Inf):=plot(Im(f1(x)),x=1..Inf,color=blue,tickmarks=[21,3],style=point):
P2(Inf):=plot(abs(f1(x)),x=1..Inf,color=red,tickmarks=[21,3],thickness=3):
display([P0(Inf),P1(Inf),P2(Inf)]);

From the red graph, it appears that abs(f1) has local minimums (mins) at every odd value of x and the value of a min is getting closer to the value of the next odd value as the value of x grows without bound .

By definition, the red graph, the absolute value of f1 is simply sqrt((-1/2*sin(n*Pi))^2+Re(f1(n))^2); in maple's natural simplification: (I say nothing esle about that.)

ff1:=sqrt((-1/2*sin(n*Pi))^2+Re(f1(n))^2);

Looking further along the x-axis of graph we see that those mins head twoard the MRB Constant, here called f .

f:=sum((-1)^n*(n^(1/n)-1),n=1..infinity):
Inf:=200:

mrb:=plot(f,x=120..Inf,color=black):
P0(Inf):=plot(Re(f1(x)),x=120..Inf,color=green):
P1(Inf):=plot(Im(f1(x)),x=120..Inf,color=blue):
P2(Inf):=plot(abs(f1(x)),x=120..Inf,color=red,thickness=2):

#3 graphs zooming in on the MRB:
display([mrb,P0(Inf),P1(Inf),P2(Inf)]);
display([mrb,P0(Inf),P1(Inf),P2(Inf)],view=[120..200,evalf(f-1/2^2)..evalf(f+1/2^2)]);
display([mrb,P0(Inf),P1(Inf),P2(Inf)],view=[180..200,evalf(f-1/2^3)..evalf(f+1/2^3)]);

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In the spirit of JacquesC’s reply, since f1:=x->sum((-1)^n*(n^(1/n)-1),n=1..x); is an absolutely converging series. The work of crunching out the digits of f1 qualifies for the following methods of (accelerated) convergence; however, practically speaking, only a few really are applicable.

http://numbers.computation.free.fr/Constants/Miscellaneous/seriesacceleration.html

More convenient are two other methods that I know of. One is mentioned in
Let.doc

Since f1 is a partially alpha-convex series(after about the third or fourth term), the methods in

Download 565_4148474.pdf
apply but are hard to understand.

On the other hand, since f1 is a reasonably well-behaved function with a(n)->0 as n->infinity, the algorithms found in

http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.em/1046889587
are easier to use.

Attached, also, is a worksheet that demonstates the absolute convergence of f1.

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n-Dimensional Cubes

Previously, we read the link QA-1.doc.
Then we looked at the worksheets describing functions f and f1;
we began to investigate how the summing of their terms can produce a value that represents the value of the MRB Constant.
In this worksheet we will use a pair of sequence of terms s and s1;
we will use them to stack n-dimensional cubes on a 2D coordinate plane in a way that also represents the value of the MRB Constant.

restart;

with(plots):

Warning, the name changecoords has been redefined

s:=n->(-1)^n*n^(1/n):s1:=n->(-1)^n*(n^(1/n)-1):alias(p=pointplot):inf:=2^10:

First we have a stacking of n-dimensional cubes:

the sequence, s, centered at the y-axis, and afterward the series, s, stacked according to QA-1.doc.

p({seq([s(n),n],n=1..inf)});
p({seq([sum(s(u),u=1..n),n],n=1..inf)});

Next, we have a stacking of n-dimensional cubes:

the sequence, s1, centered at the y-axis, afterward the series, s1, stacked according to QA-1.doc.

p({seq([s1(n),n],n=1..inf)});
p({seq([sum(s1(u),u=1..n),n],n=1..inf)});

As the graph goes up -- i.e. as y goes to infinity -- the plateau is removed and it is replaced with a peak.
We see that the sides of the two stacks are shifted from their position in the graph of s to other positions in s1.
Graphically, this peak makes the series' -- i.e. s1 and accordingly f1 -- absolutely converging.
Since we now have a peak, the top left of the map of the series s is shifted by the absolute value of the MRB Constant.

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MRB Constnat-A

MRB Constnat-A

MRB Constnat-A

See
MRB Constnat-B

MRB Constnat-B

MRB Constnat-B

We have been using a process called mit to manufacture the various family members of the MRB Constant.
Continuing in that mode we will use it again to compute m, or the MRB Constant.

restart:

mit := proc (it) global git, m; git := proc (n); (-1)^n*(n^(1/n)-it) end proc; m := sum(git(n),n = 1 .. infinity) end proc;

m:=mit(1):

This is important for understanding the following, the MRB Constant=mit(1)=g1, mit(0)=g0, mit(2)=g(2), etc

Previously when asked:

i. What real numbers to (the power of) their own roots equal mit(1)?

ii. What real number do we get when we raise mit(1) to the power of itself an infinate number of times?

We discovered:

i. and ii. have the same answer.

r:=fsolve(x^(1/x)=mit(1),x);

.

Before I show you the following trick, let's review in the following sentence.

m Is the alternating sums of all integers to (the power of) their roots; r is one number to its root that equals m.

Now we go a step further, to the area under the curve of r to (the power of) all roots. And find out -- that area is nearly rational!

int(r^(x/r),x=1..infinity);

4466/39750.;

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As we move on, this would be a good time to introduce a
process that will give us desired approximations to the MRB Constant.
I give this to you with the following caveat. Do to the internal structure of Maple
the last 2 to 4 digits may not always be correct.
However, you can use this process to compute many thousands of digits
of MRB. I hold the present world record of 40,000 digits. With little or no
fine tuning of the process, you might be able to beat my record.

e:=100:

My Desire

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It looks like we will need to do some tweaking to beet my 40,000 digit record. With the process I gave you last time, most Maple versions seem to wear out between 2000 and 5000 digits.

This first improvement is mostly cosmetic.
It uses hyperbolic functions.

for a from 2 to 20 do mrb1(a) od:

[same results as in Some Numeric Qualities Part 7 and moving on]