Dear Reader in Calculus II,
I wish there were enough time to properly get excited about sequences and series. This chapter is dynamite. Allow me to explain where you’re at. Calculus takes for granted that you have an idea of fractions of the form 1 over infinity.
This is because it assumes a strong mathematical background that is only cultivated through curiosity about the subject of mathematics. Your study of arithmetic is enough to find Sums and Series, don’t worry for your grade – you can divide by fractions and that is enough. That’s where you are and without any accusation of sloppy thinking, let’s turn our attention to convergent sequences.
The idea of convergence first appeared in the section on Improper Integrals. The integral of x^-2 with respect to x from 1 to infinity is equal to lim b->infinity -1/b+1. But this equals one, so we say the integral converges to 1.
Some of the concepts center right in on top-heavy and bottom-heavy fractions and ask you to look for a ratio between the numerator and denominator and let the variable x or n be very big and tend to infinity. You know that there are numbers bigger than 1000.
Budgets, Astronomical Units, and Integrals that involve tiny distances leave you unaffected by Big Numbers.
We will find partial sums and let the limit of the terms tend to infinity. Infinity is an idea I would like to explain in detail.
The wily Zeno motivates us to study Calculus in chapter one of your book, saying that Achilles cannot catch the turtle, but I hope that you appreciate that one half plus one fourth plus one eighth… converges to a limit – namely 1.
Please note that the partial, the sum from n=1 to n=p of (1/2)^n where n is a natural number is less than 1 – so the fractions added up are less than the whole. When we add all the parts of this 1 together, we have the whole.
To solve our problems, we will use partial sums and take the limit of the sequence of partial sums and get the series.
I hope that if you look back on chapter one, you will notice that sequences and series solve the problems that Newton and Liebnitz set out to investigate. Infinite sums are based on the understanding of the infinite and this is how it all began.
Here goes: Infinity comes from the Greeks. They were interested in the Geometry of the circle – the limit of a regular n-gon as n approaches infinity. The Greek Archimedes, author of The Sandreckoner, dispelled the notion that the number of grains of sand on the beach is infinite because look at the number of grains of sand that would fill up the Earth and then make a large sandball to encompass the mountains, it is more than the amount of grains of sand on a simple beach.
The Archimedean Property says that there is always a number one more than a given number. This idea is equivalent to the existence of infinity. This idea of more is the same as the infinite.
Let’s not be defeated. The key is “very big” and “infinite” are entirely different. The integers are sufficient for counting to 10^(10^(10^34)).
It just takes a very long time to count to this number, the number of possible things a proton could do in the universe, according to Whitehead. We measure this big number with our Natural numbers, n.
The Archimedean property is that for any number, a bigger number n+1 that is one more than the one we have exists. So there is no biggest number. There is more, that is the infinite. The infinite can be easy to understand sometimes. The infinite has a double aspect – the infinitely large and the infinitely small. Now you have the work of showing that certain limits are zero.
Here we go: What we have once done, we can do again. If a function has only one behavior, like 1/x for x>1, then it is monotonic.
If a function is never negative, then it is bounded below by zero. If a function is never over 9000 then it is bounded above by 9000.
If a function is monotonic and bounded, then it is obvious that it converges.
If a function or sequence is monotonic in its increase or decrease but it never exceeds its bounds, then the sequence does not diverge but converges to some number L.
This is the Monotonic Convergence Theorem. It is necessary to define the word monotonic with the monicker, one-minded, or doing one thing. Anyways, the real Monotone Convergence Theorem says that any bounded monotonic sequence converges. Practice your stuff.
Write into the journal if you have a question.
Liebnitz rolled up with his infinitesimals and Euler had this to say:
“There is no doubt that every quantity can be diminished until it vanishes completely and is reduced to nothing. But an infinitely small quantity is simply an evanescent (disappearing) quantity and therefore actually equal to zero.”
Then he proved that any number that is an infinitesimal is zero.
The infinitely small and the infinitely close to L, where L is the limit of a sequence, is the number that is less than E away from L, where E is any number, commonly small.
Let E>0. Suppose that there exists a number B such that for all terms after the B term, d(L, the term) < E or the distance between L and the term is less than E. E can be any number.
But Euler proved that the number less than all positive numbers is zero. Then that means d(L, the infinite term)=0, or the Limit is equal to infinite term.
You will do a great deal of counting, the best of luck.
Sincerely,
Nick
Nikolay Vasilyev
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