{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

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.

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.

 

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.   

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.