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Holden

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The one unforgivable sin is to be boring
« on: June 17, 2017, 03:26:40 am »
In this thread I'd post mathematics which maybe anything-but it will not be boring.


The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

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Re: The one unforgivable sin is to be boring
« Reply #1 on: June 17, 2017, 11:06:19 am »
So many things are going through my mind, especially considering Raul's last post about Pythagoras and his Brotherhood.  I wonder what they would have thought about Euler's Number e.  I wonder what Wildberger thinks about this "real number".

The formula is like some wizard's magic potion, no?

Then there is the representation in a computer algebra system where the symbol "e" takes on two different meanings depending on the context which it is used.  it is amazing that they are able to design such systems to know the difference.

 56e9*e^(0.025*1.75) = 5.85043839493512e10

To the uninitiated, this might cause some confusion, since 56e9 means 56 x 10^9 = 56 billion = 56000000000, whereas the 'e' in e^(r*t) is approximated as
e.n() = 2.71828182845905.

It looks much better with paper and pencil, but I doubt anyone alive today would attempt to perform the arithmetic without the assistance of an electronic calculator of some sort.

Another thing that came to mind is the political agenda behind such calculations.  In other words, there is a hidden message in such mathematical forecasts that sees an increase as "good" and a decrease as "bad".  Many years ago when I was struggling with the reality of my somehow having become one of the countless superfluous deadbeats of the Industrialized World, I read a book called "The Joy of Not Working: A Book for the Retired, Unemployed and Overworked.  I searched my abandoned blog at wordpress, but my reference to this must have been posted on the borad that got zapped in 2013 or so.  I won't mention it's url since it's name was just a strange coincidence.

I will reconstruct it here from Library Genesis (the gift that keeps on giving).

Found it: page 34-...

What the G in GNP Really Stands For


Economists, businesspeople, and politicians tell us we will all be better off if our countries have substantial increases in the gross national product (GNP). Gross national product is the value of all services and products sold in a country during any given year. It is the measurement that tells us if we have been successful as a nation. The wise men and women of business and economics tell us that the goal in any country's economy is growth in GNP.

Another goal for the economy is to eliminate unemployment. The ability to generate new jobs is dependent on economic growth. A certain level of GNP is supposed to provide jobs for everyone able to work, whether they want to work or not.

Having taught economics courses at private vocational schools and universities, I have always had a problem with GNP as a yardstick of prosperity. GNP is improved by increases in such questionable activities as consumption of cigarettes and the production of weapons. A substantial increase in car accidents will favorably affect GNP because more funerals, hospital visits, car repairs, and new car purchases will result.

With the growth in GNP considered such an important yardstick, it surprises me that the skipper on the Exxon Valdez didn't receive a Nobel Prize for economics. The gross national product in the United States increased by $1.7 billion due to the Exxon oil spill. More such massive oil spills would do wonders for the GNP. Lots more people would also be employed.

Growth in GNP for the sake of growth doesn't necessarily reflect something beneficial to society. Growth for the sake of growth is also the philosophy of cancer cells. Instead of standing for the gross national product, GNP should stand for the grossness of the national product.

We in North America can probably get by with half the resources we use and still maintain a good standard of living. This can be accomplished, in part, by changing our values. We must eliminate frivolous work and consumption, such as the production of stupid trinkets and gadgets that people buy and use for a week or two before throwing away.

More than a hundred years ago, John Stuart Mill predicted that if the world continued on its path of economic growth, the environment would be totally destroyed.


Not boring, indeed.  Who could know that a little mathematical exercise would spur such reflection?

One last thing that also came to mind in this flood of thoughts was the geometric definition of "e and the natural logarithm" I was introduced to while studying Sheldon Axler's "Algebra & Trigonometry".

(p349-357)

e is one of those things Wildberger refers to as "The Transcendentals", and I assume he would like to do without them ... anything involving infinite series ...

Myself, I don't share the same prejudice.  Even though I have a special respect for rational and natural numbers, I am not totally against e, pi, sqrt(-1), sine, cosine, tangent, etc ...

I find I am still quite impressed with even the most elementary algebra and trigonometry.  As you know, boredom and ennui is right up there with pain in Schopenhauer's conception of what it means to suffer.

The strange thing is that to many, mathematics has a reputation of being somehow boring. 

I am not going to leave any links to any definitions of euler's number but prefer that you, if it is not too much trouble, download the Axler book from libgen.io and refer to pages 352 to 354.

e is the number such that area(1/x, 1, e) = 1.

Of course, that will lead back to page 350 to see what is meant by area(a,b,c)

All in all, it is interesting if you are drawn to it at the moment, but maybe a distraction if you were studying something else.   

I have been making little breakthroughs trying to incorporate computer algebra systems and graphing calculators into my studies. 

Yesterday I was kind of surprised that some trouble I was having representing (graphically) systems of inequalities using Sage were actually helped when I incorporated the old-fashioned method of constructing a graph using a table of values.  This seems to take too much time, but it can be automated by using a TI-Nspire.  I won't go into details, but when I took the extreme ends of my values for x which I plotted by hand, and used these as the arguments in the parameters for the plot in SageMath notebook(), I made a breakthrough.

There was not much documentation for this simple task that can be accomplished by hand by shading in regions with a couple different colored pencils.  It was more work to get Sage to do it, but it forced me to get a feel for my own way of making it happen, and, in the end I was happy with how the Pythonese code resembled the mathese.  I will post a snippet to show you what I mean.  I love when I can do something with computers where the code I write looks unmistakeningly familiar to its mathematical counterparts.

If I could just figure out a way to reproduce it here, you could witness how UNBORING (I know this is not a word) even some seemingly boring math can be.

It can be "technically challenging" to find a balance between the old ways and the so-called modern or technologically-assisted ways.

If one does not first do these things by hand to get a feel for the whereabouts of the graph in the plane, then the code generated graph will not look right, if you can get the code to run at all.  When you get it right, there is a sense of fulfillment, but it is on a very personal level and nothing to brag about since one is always taking baby steps.   

Maybe in the future we could use sage math cloud which has already been rebranded.  I mean, this way you would not have to get Sage up and running on a mission or virtual machine. Eventually we will be able to copy from here and paste into a live Sage session, so I'll leave it here for now.

The gist:  I was quietly thrilled with this little breakthrough which occurred rather spontaneously yesterday while "farting around"/"tinkering" with "sage notebook()" :


x = var('x')
f(x) = x^2 - 2*x -8
g(x) = 8 - 2*x - x^2

def h(x,y):
    return (y > x^2 - 2*x - 8) & (y <= 8 - 2*x - x^2) # goddamn emoticon is an 8 and )

# This quite evilly mocks my feeble attempt to communicate how math-like the code looks ::)
#  In "mathese" :  {(x,y): y > x^2 - 2*x - 8} ⋂ {(x,y): y <= 8 - 2*x - x^2}
   
G = Graphics()
G += plot(f, -4,6, linestyle="--")
G += plot(g, -6, 4, linestyle="-")
G += region_plot(h, (-10, 10), (-9, 12), plot_points = 300, incol='gold')
G.show()



f(x) = 3 - 2*x - x^2

def g(x,y):
    return (y < 3 - 2*x - x^2) & (2 >= abs(y))
   
   
G = Graphics()
G += plot(f, -5, 3, linestyle="--")
G += plot(2, -10, 10, linestyle="-")
G += plot(-2, -10, 10, linestyle="-")
G += region_plot(g, (-10,10),(-10,10), plot_points=300, incol='green')
G.show()


_____________________________________
PS:

In SymPy, although the graph is not as elaborate, for the second plot, one line suffices, which is kind of "elegant":

p = plot_implicit(And(y<3-2*x-x**2, 2>=abs(y)))

 :-\

« Last Edit: June 18, 2017, 01:44:03 pm by Raskolnikov »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Breakthrough? sagecell.sagemath.org/
« Reply #2 on: June 18, 2017, 01:53:34 pm »
You can copy the following and paste it into a cell at sagecell.sagemath.org!  This might help us to communicate. 

Don't worry about the emoticon.  It will magically disappear when you paste the code into the cell.  Do this for each code snippet separately.
--------------------------------------------------------------------------------------

x = var('x')
f(x) = x^2 - 2*x -8
g(x) = 8 - 2*x - x^2

def h(x,y):
    return (y > x^2 - 2*x - 8) & (y <= 8 - 2*x - x^2) # goddamn emoticon is an 8 and )


#  In "mathese" :  {(x,y): y > x^2 - 2*x - 8} ⋂ {(x,y): y <= 8 - 2*x - x^2}
   
G = Graphics()
G += plot(f, -4,6, linestyle="--")
G += plot(g, -6, 4, linestyle="-")
G += region_plot(h, (-10, 10), (-9, 12), plot_points = 300, incol='gold')
G.show()
____________________________________________________________


f(x) = 3 - 2*x - x^2

def g(x,y):
    return (y < 3 - 2*x - x^2) & (2 >= abs(y))
   
   
G = Graphics()
G += plot(f, -5, 3, linestyle="--")
G += plot(2, -10, 10, linestyle="-")
G += plot(-2, -10, 10, linestyle="-")
G += region_plot(g, (-10,10),(-10,10), plot_points=300, incol='green')
G.show()

Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: The one unforgivable sin is to be boring
« Reply #3 on: July 30, 2017, 02:10:13 pm »
Holden,

You once expressed interest in linear programming.

I wonder if this sage code would work in the sagecell referenced to above.

I placed the following in a sagecell, and the result was different than what it was on the resource page.

p=MixedIntegerLinearProgram( maximization=True )
p.set_objective( 2*p[1]+p[2] )
p.add_constraint( 3*p[1]+4*p[2], max = 2.5 )
p.add_constraint( 1.5*p[1]+0.5*p[2], min = 0.5, max = 4 )
p.solve()
print "The optimal values are x_1 = "+str(p.get_values(p[1]))+", x_2 = "+str(p.get_values(p[2]))

I get 4.722222222222221 while they said it would be  1.6666666666666665.

Also, whereas they show:   The optimal values are x_1 = 0.833333333333, x_2 = 0.0

My result was:  The optimal values are x_1 = 3.27777777778, x_2 = -1.83333333333

buggy?

The next one we were in agreement with:

g=graphs.PetersenGraph()
p=MixedIntegerLinearProgram(maximization=False)
b=p.new_variable()
for (u,v) in g.edges(labels=None):
    p.add_constraint(b[u]+b[v],min=1)
    p.set_objective(sum([b[v] for v in g]))
    p.set_binary(b)
    p.solve()

b = p.get_values(b)
print b

print [v for v in g if b[v]==1]

[/u]
Anyway, just a reminder that, even though I have been obsessed with installing SageMath in every operating system, you can always experiment a little with the online sagecell.

Consider this "future reference".   I understand if this is not your thing at the moment.

If you ever decide to check out Sage (or anyone else who passes through here by accident), there is a good essay on Sage on a CUNY Math Blog

Sage Tutorial for linear programming

Also see Sapien Games:  History, Politics, & Other Heresies
« Last Edit: March 29, 2020, 11:40:42 am by mike »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Re: The one unforgivable sin is to be boring
« Reply #4 on: July 31, 2017, 09:46:30 pm »
I understand that you are not currently interested in anything related to programming.  I also realize that Linear Programming does not necessarily have anything to do with computer programming, and that it finding minimum and maximum values given constraints on a system of linear equations.

I only placed these links here like someone placing a pair of warm socks in the back of a drawer for some distant season.

Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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To Herr Hentrich
« Reply #5 on: July 31, 2017, 11:39:10 pm »
Herr Hentrich,
I am very sorry for my late response.I am travelling. I do greatly appreciate your posts about maths.

Once again I must say that if not for our preordained meeting in 2014 I might still be drifting away aimlessly.
It is only because of you that I possess some sort of sense of direction.I am studying mathematics but it is of very fundamental kind.If my foundations are weak that I can never expect to study math of more advance variety. I am studying things like :the infinite sum of 1+4/7+9/7^2+16/7^3+25/7^4+.......

I am taking it very slowly but I am STUDYING maths.I am not saying this just to please you.I am genuinely interested in maths.

By the way,did you know that Kant was a very competent mathematician and was very much interested in Newtonian mathematics?
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

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Re: The one unforgivable sin is to be boring
« Reply #6 on: August 01, 2017, 01:07:53 pm »
That's good to hear, Holden.  I share your views about the importance of the fundamentals, which is why I have temporarily put aside the texts on multivariable calculus, linear algebra, differential equations, physics, etc, and am challenging myself to take seriously a couple high school geometry texts.  I thought I would breeze through them over the summer, but I suspect I will be "stuck" in Geometry Land for a good year before I am able to continue with my "Algebra/Trigonometry/Analysis" coding.

I had wanted to go through the Dolciani series, translating the programming exercises from BASIC and Pascal to C/C++ and even SAGE/Python, but I found the Dolciani series weak in the subject of Geometry so I am incorporating a high school (honors) textbook from 2011 that I found on ebay for $13 (to supplement the more formal and traditional Jurgensen/Brown/Jurgensen text).   

While it does contain some goofy looking characters that at first I found annoying, some of the "challenging" exercises are exactly that.  I wake up in the morning and try not to classify myself by age or (lack of) occupation.  Fortunately I am not under anyone's microscope so I am able to engage with the material very HONESTLY.  This is only possible in solitude.  There is no pressure to delude myself.  Nor is there any reason to feel ashamed if I actually have to think or if I find I am learning.

You had once told me that this is a courageous decision.  I now understand what you meant.  It takes courage to face the fact that I could use an overhaul in my understanding.  It doesn't matter that I graduated from a university 15 years ago.  I feel this need to revisit high school mathematics in order to remain true to myself. 

I have to say that I appreciate your focusing on the fundamentals for this shows me that you sincerely respect my decision to do the same.

We, Holden, are not fuucking around.  We do not aim to impress others, but are actually genuinely interested in developing our understanding.

Quote from: Holden
I am studying things like :the infinite sum of 1+4/7+9/7^2+16/7^3+25/7^4+.......

Ah ... strangely enough, I spent a good part of this past winter going over Infinite Sequences and Series.

I regret that we are not able to exchange notes with paper and pencil.  This message board makes for a very clumsy medium.

There is a way to work around the limitations of our present "editor".

If we agree on some way of making certain details clear ...

For instance, with  1+4/7+9/7^2+16/7^3+25/7^4+ ...

Would it be wrong for me to assume the following?

1 + (4/7) + 9/(7^2) + 16/(7^3) + 25/(7^4) + ....

=

1 + (2^2)*(1/7) + (3^2)*(1/7)^2 + (4^2)*(1/7)^3 + (5^2)*(1/7)^4 + ...

So, I recognize this as a geometric series, that is, a geometric sequence of partial sums {n^2 * (1/7)^(n-1)}.


When n = 1, it looks as though (1/7) is being raised to the 0 power, which gives (1/7)^0 = 1; hence the first term is 1.  We now see that (1/7) [= r] is being raised to the power (n-1).

Hence, the second term, where n = 2, is (n^2)*[ (1/7)^(n-1) ], or

(2^2)*[ (1/7)^1 ] = 4/7

It helps to break it down like this just to make sense of each term (before considering the infinite sequence).

So the third term has n = 3 and n-1 = 2:

(3^2)*[ (1/7)^(3-1) ] = 9*[(1/7)^2] = 9/(7^2)

The fourth term:  (4^2)*[ (1/7)^3 ] = 16/(7^3)

and so on.

The main thing is to recognize and extract the form.

You want to be able to explain this to a computer algebra system, something like, where "sum" represents sigma or summation:

sum(n^2 * (1/7)^(n-1), n, 1, oo)

"The sum of n squared times (1/7) to the (n-1) power for n from 1 to infinity."

Input n^2*(1/7)^(n-1) over here for n = 1 to infinity.

The sum is 49/27.

To understand why this is, think about the limit.  The terms of this series converge to 1/7, which is less than 1.  When considering the limit, notice that {n^2 * (1/7)^(n-1)} will be called a_n, where n is the subscript.

a_subscript_(n+1) is just {(n+1)^2 * (1/7)^n}

In the following, they check the limit (as n approaches infinity) of the ratio a_(n+1) / a_n:

See here

The reason I am utilizing this "calculator" sites is because of the mathematical notation they are able to employ.

This helps us communicate and have some confidence that we are on the same page, so to speak.

For now, you probably don't need to think about limits (the second link) and are more concerned with finding the sum of the series.

This brings attention to a little dilemma. 

Of course, there must be some kind of formula, right?

If it were not for the n^2 part, it would be straight-forward:

1/(1-(1/7)) = 1/(6/7) = 7/6

If you ask a CAS to solve: sum((1/7)^(n-1), n, 1, oo), you will also get 7/6.

The sum of n^2 from 1 to infinity is infinity, but the combination of the two gives a sum of 49/27 ... so there is a factor of (7*2)/9 in there somewhere. 

It's amazing that I was covering similar material throughout the winter, and I have to refer to my "notes" or else risk adding to the confusion rather than clarifying anything. 

This is both funny and sad at the same time, no? 

I would suggest looking at a problem of the form a + a*r + a*r^2 + a*r^3 + ...

The n^2 term is adding a level of complexity to the problem.

Start with 1 + 1/7 + (1/7)^2 + (1/7)^3 + ... first.

That's my suggestion.  Anything involving limits and infinity is begging to use calculus.  At least with the standard geometric series, you can apply some formulas.

Please do not hesitate to also check results with a CAS as this will help verify the results you get with a formula.

The reason for the formulas is that without them, you would require calculus.

Don't get discouraged.  Meanwhile I have to eat some food before my brain becomes totally useless.

That's why the most important thing in all the schools on the planet would be to offer breakfast to all the students.  First class:  HOME GROUP, then some kind of breakfast!  Without some nutrition, all the teaching is in vain.

I have been up since morning time, it is after 2PM, and I have not eaten yet.   So, it is no surprise that my brain is fuzzy and I'm a little dizzy.

 :P

It will be interesting to see if it turns out that we both have a low frustration tolerance when it comes to mathematics.  I have found that, no matter what the "level" of mathematics I am looking at, the threshold for when I lose patience with myself is small.  This is not the case in a classroom situation where I become engrossed in the task at hand, but when I return to anything outside that particular type of math, it is as though I am starting all over again.  This leads me to suspect that I may delude myself into thinking I have comprehended something.

I do not say this to discourage you, but quite the opposite.  I don't want to project an image of someone who has a deep understanding of things, but rather, I want you to feel more comfortable with your own frustrations upon witnessing how much concentration I have to harness just to get my bearings.   

In fact, I am moving in a direction where I will either embrace the total disintegration of my ego or be hanging from the end of a rope.   :D

I have to give my brain permission to be itself and not to bombard it with demands about "what it ought to understand".  It is what it is, and I have to just get over the fact that my brain is not a storage bin for masses amount of knowledge, but rather a chaotic bundle of nerves which can only focus on the task at hand.

Since I do have a rather sensitive ego, isolation and privacy are crucial to such undertakings.  I seek guidance from others (textbooks and other sources), but it has to be at my own slow pace. 

Maybe we are unwittingly engaged in what could fall under the title "The Politics of Mathematical Empowerment".

Too much of formal education tends to have the opposite effect of mathematical disempowerment, hence, in her suicide note Kriti Tripathi has urged the government of India and human resource development (HRD) ministry to shut coaching institutes as soon as possible. “They suck,” she wrote.

Footnote (originally linked to by Holden)

There has to be a way to make this learning process more individualized so that we might appreciate the little we can cover in one lifetime rather than feeling that there is no point in it.

As you say, we are concerned with developing a solid foundation.  I would add that the foundation and fundamentals are a worthy area of study in and of itself, and that there need not be any distant destination.

We are, after all, merely doing time in the penal colony of existence.  This may be a good way to get through the days and years and decades.
« Last Edit: August 01, 2017, 06:18:44 pm by { { } } »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: The one unforgivable sin is to be boring
« Reply #7 on: August 01, 2017, 11:53:03 pm »
Herr Hentrich thank you so much for solving the problem.Your solution is correct.
I have a little algorithm which I make use of to solve such problems. Here is how it goes:
S=1+4/7+9/(7^2)+16/(7^3)+25/(7^4)+....

Now what I do is ,I multiply both sides with1/7:
1/7 S=1/7+4/(7^2)+9/(7^3)+16/(7^4)+...
Now,I substract the 2nd equation with the first:
S (6/7)=1+3/7+5/(7^2)+7/ (7^3)+....

Again,I multiply both sides with 1/7:

 S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....
Now ,I substract the 4th equation from the third:
S (36/49)=1+2/7+2/(7^2)+....

This is a geometric progression with first term=1 and common ratio=1/7
=1+(2/7)(1/(1-1/7)
S=49/27.
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

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Re: The one unforgivable sin is to be boring
« Reply #8 on: August 02, 2017, 12:32:58 am »
Let me see if I am following your logic.

Subtracting the 4th equation from the third:

S (6/7)=1+3/7+5/(7^2)+7/ (7^3)+....
 S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....
---------------------------------------------
S*[ (6*7)/(7^2) ]=1+3/7+5/(7^2)+7/ (7^3)+....
 S* 6/(7^2)=1/7+3/(7^2)+5/(7^3)+.....
----------------------------------------------------------

S*[ (42-6)/7^2 ] =1 + [3/7 - 1/7] + [ (5-2)/(7^2) ] + [ (7-5)/(7^3) ] + ...
-------------------------------------------------------------------

S (36/49)=1+2/7+2/(7^2)+....

_____________________________________________
OK, then what?  Let's see ... Do you then factor out the 2? or are you factoring out 2/7?

S*(36/49) = 1 + (2/7)*[1 + 1/7 + 1/7^2 + 1/7^3 + ...]

ah, and that is your geometric progression with first term=1 and common ratio=1/7 ... where you use the formula 1/(1-r) ---> 1/(1-(1/7)) ---> 1/(6/7) ---> 7/6

So, S*(36/49) = 1 + (2/7)*(7/6)
S*(36/49) = 4/3
S = (4/3)*(49/36) = (1/3)*(49/9) = 49/27

Very clever.  Kind of Gaussian.  And no computers, you rebel!  Sorry if I doubted you for a minute.  I had to prove it to myself.  Wow.  That's very cool, Holden.

I like that.  I was just shutting down the computer for the night, so I checked the board first.  I'm pecking away at Infinite Jest again.

That is a very cool method you used, a real confidence builder where you attack it with brute force. 

Your method reminds me a little of how one finds the sum of numbers between number x and number y, where you have

S = x + (x+1) + (x+2) + ... + (y-2) + (y-1) + y

Then you add the same sum (in reverse order):

S = x + (x+1) + (x+2) + ... + (y-2) + (y-1) + y
S = y + (y-1) + (y-2) + ... + (x+2) + (x+1) + x
--------------------------------------------------------
2S = (x+y) + (x+y) + (x+y) + ... + (x+y) + (x+y) + (x+y)

There are n terms, so 2S = n(x+y)

Then, dividing both sides by 2, S = [ n (x+y) ]/2

Thanks for the inspiration.  It's good to be able to work such things out by hand without a full scale 8 gigabyte computer algebra system!   :D


Post Scriptum:

If you ever want to check your results, and you can extract {a(n)} from the given terms of the series, go to http://sagecell.sagemath.org/.   For example, copy the following into the cell and press evaluate:

n = var('n')
h(n) = (n^2)*(1/7)^(n-1)
sum(h(n), n, 1, oo)

Note that oo represents "infinity", that is, two lowercase o.
« Last Edit: August 02, 2017, 03:42:29 pm by { { } } »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

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Holden

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Re: The one unforgivable sin is to be boring
« Reply #9 on: August 02, 2017, 10:25:30 pm »
Thank you for the response. I have just reached home from the tour and am tired.Will post again in the evening.
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

Holden

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Binomial Theorem
« Reply #10 on: August 04, 2017, 01:46:47 pm »
Herr Hentrich,very sorry about the late reply.I have been reading a bit about the Binomial Theorem. Its a tricky one. Here is a video you might like to check out sometime:    https://youtu.be/OMr9ZF1jgNc
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

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Re: The one unforgivable sin is to be boring
« Reply #11 on: August 06, 2017, 12:56:41 pm »
Herr Hentrich, I have found that here the maths books have a lot of misprints,much more than any run of the mill book. Also, that it is very important to identify when a maths problem has been wrongly elucidated otherwise one ends up losing confidence in one's understanding because of some lazy book printer.Well, any way just to give you the idea about the kind of things I am grappling with at the moment-"All the page numbers from a book are added,beginning at page 1 .However, one page number was mistakenly added twice.The sum obtained was 1000. Which page number was added twice? Some questions are very badly written. But I am trying to sort out these things. Keep well.
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.

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Re: The one unforgivable sin is to be boring
« Reply #12 on: August 06, 2017, 01:01:36 pm »
We are in synch, Holden.

In the video above, the guy stops short just before he had the opportunity to introduce a formula based on Pascal's Triangle.

In the following video, you will have to forgive the instructor for his sloppy hand writing and sometimes quirky way of speaking (as in, using the acronym "FOIL" as a verb and other idiosyncratic annoyances).  At first, the voice may conjure up images of the overbearing Hasidic Jew in the black and white film, PI, but you can tell he really loves math.

Unlike that character in PI, this guy eventually grows on you, becoming even amiable.  I especially like his genuine appreciation of Sage.  He is refreshingly enthusiastic about it.  This reinvigorates my interest in computer algebra systems.

What attracted me to his videos in the first place is I was curious to see how he would present Sage to high school students.

Strangely, I was viewing this video when you sent the above link.

Can I use the word "uncanny" here, or is the term "coincidence" more appropriate?

Skip through to about 7 minutes.

You are interested in patterns, and there is a cool pattern which includes the row and column of Pascal's triangle without actually having to write the entire triangle of numbers.  Probably in around 11 or 12 minutes he gets to the good shiit, Combinatorics:  (nCr).

Each row is identified by n, where the first row of Pascal's Triangle is n = 0, the second is n =1, etc, and each column "r" is r = 0 for first column, r =1 for second column, etc .

For example, 3C2 would correspond to the fourth row, third column.

You can find each coefficient in this manner.

All in all, there is no getting around the tedious drudgery involved ... which is why one can't help but appreciate Sage (free, as in free beer).  Observe how elegantly it constructs the following:

Copy this into a sagecell:

for n in range(10):
    show(expand((x+1)^n))

Note that, if the SageCell complains about a syntax error, it is very picky about spaces.  Backspace from show until it is just before the colon, :, and then hit enter so show(expand((x+1)^n)) is indented precisely with the s under the n.  That is, four spaces indented.

shadowfax

Maybe skip into 26 minutes at first.  You have to patient as the video is in real time, which means the instructor is assisting students with "technical difficulties" throughout the video.  The example is (3*x + 1)^4, and he shows the coefficients in terms of nCr, where nCr --->  (n   r)  [but vertical instead of horizontal, obviously]

nCr means (n!)/[r! * (n - r)!]

So, 5C2 means (5!)/[2! * 3!] = (5*4)/2 = 10

Notice the 3! = 3*2*1 in the denominator cancels out the 3*2*1 in 5! leaving 5*4 in the numerator.


----------------------------------------------------------------------------------
PS:  for (3*x + 1)^4
In Sage:

for n in range(4):
    show(expand((3*x+1)^n))

def combo(n,r):
    return factorial(n)/factorial(r)/factorial(n-r)

I like the way he divides twice instead of typing the denominator with parentheses and using multiplication:  factorial(n)/(factorial(r)*factorial(n-r))

As he points out, multiplication by a reciprocal is division, so it is cooler, I think to write it as double division ...

Cheap thrills, I know.   8)
---------------------------------------------------------------------------------------

On paper, how would you show the work?  What's the pattern?
Let nCr be represented by the defined function, combo(n, r)
To find binomial expansion of (3*x + 1)^4:

    combo(4,0) * (3*x)^4 * (1)^0
 + combo(4,1) * (3*x)^3 * (1)^1
 + combo(4,2) * (3*x)^2 * (1)^2
 + combo(4,3) * (3*x)^1 * (1)^3
 + combo(4,4) * (3*x)^0 * (1)^4

 = [4!/(0!*4!)] * 81*x^4 * 1
 + [4!/(1!*3!)] * 27*x^3 * 1
 + [4!/(2!*2!)] * 9*x^2 * 1
 + [4!/(3!*1!)] * 3*x * 1
 + [4!/(4!*0!)] * 1 * 1
 = 81*x^4 + 4 * 27*x^3 + 6 * 9 * x^2 + 4 * 3*x + 1
 = 81*x^4 + 108*x^3 + 54*x^2 + 12*x + 1

In sage:
combo(4,0) * (3*x)^4 * (1)^0 + combo(4,1) * (3*x)^3 * (1)^1 + combo(4,2) * (3*x)^2 * (1)^2 + combo(4,3) * (3*x)^1 * (1)^3 + combo(4,4) * (3*x)^0 * (1)^4

In a sagecell, copy and paste the following:

def combo(n,r):
    return factorial(n)/factorial(r)/factorial(n-r)

show( combo(4,0) * (3*x)^4 * (1)^0 + combo(4,1) * (3*x)^3 * (1)^1 + combo(4,2) * (3*x)^2 * (1)^2 + combo(4,3) * (3*x)^1 * (1)^3 + combo(4,4) * (3*x)^0 * (1)^4)


The pattern is

for r in range(5):
    combo(4,r) * (3*x)^(4 - r) * (1)^r

So this is a series (a + b)^n = Sum( combo(n,r) * a^(n - r) * b^r, r, 0, n)

The sum of combo(n, r) * a^(n - r) * b^r from r = 0 to r = n

I will take it one step further than MrG in the above video, and write the summation as sage code. Remember we have to define combo(n, r) and the variable r in the sagecell.

In Sage (for the binomial expansion of (3*x + 1)^4), where a = 3*x and b = 1 [n = 4]:

def combo(n,r):
    return factorial(n)/(factorial(r)*factorial(n-r))

r = var('r')
sum(combo(4,r) * (3*x)^(4 - r) * 1^r, r, 0, 4)


Copy this into a sagecell
« Last Edit: August 06, 2017, 09:36:58 pm by { { } } »
Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

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Re: The one unforgivable sin is to be boring
« Reply #13 on: August 06, 2017, 01:12:39 pm »
Quote from: Holden
Some questions are very badly written. But I am trying to sort out these things. Keep well.

Yes, that can be very frustrating, and it takes a great deal of confidence to determine just when there is such a mistake.

Now, "All the page numbers from a book are added,beginning at page 1 .However, one page number was mistakenly added twice.The sum obtained was 1000."

I would not know where to begin except brute force.

The sum of the numbers 1 to 44 is 990.

(1 + 44)/2 * 44 = 990

Page 10 may have been counted twice, which would result in 1000.

Things They Will Never Tell YouArthur Schopenhauer has been the most radical and defiant of all troublemakers.

Gorticide @ Nothing that is so, is so DOT edu

~ Tabak und Kaffee Süchtigen ~

Holden

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Re: The one unforgivable sin is to be boring
« Reply #14 on: August 06, 2017, 10:35:17 pm »
Thanks a lot for solving the book page numbers question.I could not comprehend how it was solved in the text book.You depicted it in a very clear fashion and I could get it immediately.  :)
I never thought that I should try adding up the numbers so that the sum could be as close to 1000 as possible.
I am working on the Binomial theorem in a number of ways and would share the details with you.Thanks for your input.It would help a great deal.
The more I am spent, ill, a broken pitcher, by so much more am I an artist - a creative artist - in this great renaissance of art of which we speak. ... -van Gogh.