## 550 Reputation

18 years, 216 days

I've been using Maple since 1997 or so.

## Simply enter solve(1-sqrt(3)-sqrt(2)-2*x...

Simply enter solve(1-sqrt(3)-sqrt(2)-2*x^2, x);

## It would help if we could determin...

It would help if we could determine if, in the bottom right face of the cuboid frame, are those spirals or concentric circles inside of what looks like tires. Also I would like confirmation to my assumption that this is more than one equation graphed together[like z=Arccsh(x*I+y) and arccos(z*y+tan(z-y))=sinch(y*i+x^z)], or does all of this come from one equation[like z/(x+y)^(x/(Pi*y)=arccsh(x*I+log[z](y))+1/(z+x-y)*arccossinc(y*i+x)]?

## ...

From the right side, looking left, it looks like it is a 3D contour plot.

Notice the flat top of the surface; that could be from the z range being to small to include the top of the figure. I dislike saying it; but it looks like the top surface has been smoothed, as a visual effect.

## Like nonmonotonic...

For one thing we can compare the lower right with the upper left. They are not exactly symmetric but they are close to being so. Perhaps one is for positive domain and the other for negative domain of a function that treats positive numbers only slightly different than negative ones. That would be unlike the abs operation which treats positive and negative input diametrically. However the equation responsible for the graph could be taking the abs of an expression that gives one value for some positive inputs and a different value for negative inputs, like how abs(x-7) is nonmonotonic for some range of positive input and monotonic+ for other positive input and monotonic- for all negative input.

## I sure have made the formula for...

I sure have made the formula for the MRB constant appear quite different! It is interesting to note that except for convergence, the final form of the MRB constant has the same long-term properties of the original form, Limit(sum((-1)^n*n^(1/n),n=1..2*N),N=infinity).

A triple sum for the MRB constant

 >

Below we extend notion of sum so that a series which does not converge is still subject to our analysis..

Using the theorem that says

Let t=(Pi*n^2-I*ln(n))/n), and then MRB constant = sum(sum((I*t)^k/k!,k = 0 .. infinity) ,n=1..infinity) +1/2

Not only will we formulate the MRB constant so there are no fractional powers, but we will write it so that there are no logarithmic functions either.

t=(Pi*n^2-I*ln(n))/n and

m=sum(exp(I*t), n = 1 .. infinity)+1/2.

 >
 (1)
 >
 (2)

We expanded exp(I*t) into a series  to get sum(I^k*t^k/factorial(k), k = 0 .. infinity).

Thus m=sum(Sum(I^k*t^k/factorial(k), k = 0 .. infinity), n = 1 .. infinity)+1/2

 >
 (3)

Now we have t=(Pi*n^2-I*ln(n))/n and

we can expand ln(n) into the series sum((-1)^(k+1)/k*(n-1)^k,k=1..infinity).

Thus t=(Pi*n^2-I*sum((-1)^(j+1)/j*(n-1)^j,j=1..infinity))/n.

 >
 (4)

Putting the expanded version of t into the formula for m gives

 >
 (5)

Distributing the I we get

 >
 (6)

Since I*(-1)^(j+1) = -I*(-1)^j, we have

 >
 (7)
 >

Distributing the 1/n we get

 >
 >
 (8)
 >

=

 >
 (9)

Since I and the innermost series are both raised to the kth power, we can combine like powers and get

 >
 (10)

Since -(I^2) =1 we have

 >
 (11)

Moving the innermost series to the left gives us

 >
 >
 (12)
 >
 >

Note: if you switch the sign of I*Pi*n you still get the same numeric result.

 >
 (13)
 >
 >
 >

marvinrayburns.com

## No fractional powers, per say...

When computing the MRB constant,(sum((-1)^n*(n^(1/n)-1),n-1..infinity), the hardest part is the computation of all of the fractional powers i.e. (n^(1/n)), or even the exp(ln(n)/n).

The following form is by no means optimal, but by using series expansion, it does work around the need to compute the (n^(1/n))'s.

Theorem:
Let t=(Pi*n^2-I*ln(n))/n), and then MRB constant = sum(sum((I*t)^k/k!,k = 0 .. infinity) ,n=1..infinity) +1/2
Proof:
OnFri, January 14, 2011 3:56:34 PM I showed that MRB constant = sum(exp(I*t),n=1..infinity) +1/2, where t=(Pi*n^2-I*ln(n))/n).
Now consider that exp(I*t) =sum(I^k/k!*t^k,k = 0 .. infinity)=sum(sum((I*t)^k/k!,k = 0 .. infinity) ,n=1..infinity), and so
MRB constant = sum(Sum((I*t)^k/k!,k = 0 .. infinity) ,n=1..infinity) +1/2, where t=(Pi*n^2-I*ln(n))/n).

## A mysterious form for a common series pa...

I thought I would mention another form of the MRB constant that I found. Here is a copy of my first mention of it.

From: Marvin To: Steven; Roland ; Carl ; zshen; Eric Cc: simon; Richard; psebah; support; Wolfram|Alpha Sent: Thu, January 13, 2011 6:41:45 PM Subject: Another form for the MRB constant

New Message ----

More esteemed colleagues, here are some second generation ponderings of Marvin Ray Burns.

I reshaped this new form for the MRB constant a little further in the right direction, I hope.

Remember, i am extending the notion of sum so that a series which does not converge may still have a well-defined Cesàro sum.

Lemma:

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

Proof: The MRB constant

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

= limit(sum((-1)^n*n^(1/n),n=1..2*N),N=infinity)

= sum((-1)^n*(n^(1/n)),n=1..infinity)+1/2.

QED

Now I will state a therom; the MRB constant = sum(exp(I*t),n=1..infinity) +1/2,

where t=(Pi*n^2-I*ln(n))/n).

Proof:

We know that

(-1)^n = exp(I*Pi*n),

and

n^(1/n) = exp(ln(n)/n).

Thus

exp(I*Pi*n)*exp(ln(n)/n)

=exp(I*Pi*n+ln(n)/n)

=exp((I*Pi*n^2+ln(n))/n).

Now also notice that

(I*Pi*n^2+ln(n))/n = I*(Pi*n^2-I*ln(n))/n.

Consequently,

(-1)^n*n^(1/n) = exp(I*(Pi*n^2-I*ln(n))/n)

= exp(I*t) where t=(Pi*n^2-I*ln(n))/n,

and so

sum((-1)^n*n^(1/n),n=1..infinity) = sum(exp(I*t),n=1..infinity), where t=(Pi*n^2-I*ln(n))/n.

Therefore, by the lemma, the MRB constant = sum(exp(I*t),n=1..infinity) +1/2, where t=(Pi*n^2-I*ln(n))/n).

QED

With the arrangement sum(e^(i*t)), I hope to come closer to finding a closed form for the MRB constant.

Continued best wishes to all!

Marvin Ray Burns

Original investigator of the MRB constant

## Since we have the series1/2 s...

Since we have the series

1/2 sum((exp(w+I*m)+exp(w-I*m)),n=1 ..infinity) where m=Pi*n and w = ln(n)/n.

We can see from WolframAlpha that exp(w+I*m)+exp(w-I*m) =2 e^w cos(m)

Thus now our series looks like

(1/2)*sum(2*exp(w)*cos(m),n=1..infinity)                 where m=Pi*n and w = ln(n)/n.

= sum(exp(w)*cos(m),n=1..infinity) .

And of course exp(w) =exp(ln(n)/n) = n^(1/n).

Then we have

sum(n^(1/n)*cos(m),n=1..infinity), and

cos(m) = cos(Pi*n) where

cos(Pi*n) = (-1)^n.

That gives us

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

=

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

The limsup of the function of partial sums of even n can be represented by

limit(sum((-1)^n*n^(1/n),n=1..2*N),N=infinity).

Also we know limit(sum((-1)^n*n^(1/n),n=1..2*N),N=infinity)

=sum((-1)^n*(n^(1/n)-1),n=1..infinity) which equals

0.1878596425...., which number we know.

## In  fact since we have1/2 sum((exp(...

In  fact since we have

1/2 sum((exp(w+I*m)+exp(w-I*m)),n=1 ..infinity) where m=Pi*n and w = ln(n)/n,

we can simplify this series to

1/2 sum((exp(w+x)+exp(w-x)),n=1 ..infinity) where x=I*Pi*n and w = ln(n)/n.

According to Wolfram Alpha I*Pi*n = 2 e^w cosh(x), so if Alpha is correct this series looks like

1/2 sum((exp(w+x)+exp(w-x)),n=1 ..infinity) where x=2*e^w cosh(x) and w = ln(n)/n, but I think with the 2*e^w cosh(x), it is too comlicated.

## Can take that out of the formula...

For integer n, since sin(n*Pi)=0, Isin(n*Pi)=0; so we can take that out of the formula for the series.

Let m=Pi*n and w = ln(n)/n. Then

sum((cos(m)+I*sin(m))*(cosh(w)+sinh(w)),n=1 ..infinity)

sum(cos(m)*(cosh(w)+sinh(w)),n=1 ..infinity); and

sum(cos(m)*(cosh(w)+sinh(w)),n=1 ..infinity)

=

sum(cos(m)*cosh(w)+cos(m)*sinh(w),n=1 ..infinity).

Then we can enter convert(cos(m)*cosh(w)+cos(m)*sinh(w), exp) into Maple and get

(1/2)*exp(w+I*m)+(1/2)*exp(w-I*m), which equals

1/2*(exp(w+I*m)+exp(w-I*m)).

So now the formula for our series looks like

1/2 sum((exp(w+I*m)+exp(w-I*m)),n=1 ..infinity) where m=Pi*n and w = ln(n)/n.

## The Following might Help...

I havent tried any of these ideas, but the following might help.

http://www.mapleprimes.com/posts/39944-Another-Way-To-Plot-The-Region-Between-Two-Curves

I just found this by using the search box at the top of the page.

## The easy way...

The easy way is to use the ?sum command.

restart; sum(2*n+7, n = 100 .. 100);

restart; sum(2*n+7, n = 1 .. 100);

marvinrayburns.com

## apostrophes...

Dear Rob,

M := Matrix([[1/2, 1], [1, 1/2]]); '1/2'*evalm(M)

will do it. The apostrophes keep the from being evaluated.

Best wishes,

Marvin

## Rahim, I found an example in Maple Prime...

Rahim,

I found an example in Maple Primes' history, it is at http://www.mapleprimes.com/questions/38532-How-Do-I-Plot-Points-On-The-Number-Line.

Here is what it looks like.

I hope this helps!

marvinrayburns.com

## URL...

The technical answer is found in http://www.maplesoft.com/documentation_center/maple12/Install.html#System_Requirements. Those requirements, I assume, include using the more memory intensive Document mode, so I figure that the classic interface would, more so, run OK.

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