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2017 | OriginalPaper | Chapter

3. Infinite Sequences of Real and Complex Numbers

Author : Paul Loya

Published in: Amazing and Aesthetic Aspects of Analysis

Publisher: Springer New York

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Notable enough, however, are the controversies over the series $1 - 1 + 1 - 1 + 1 -$.

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previous chapter Numbers, Numbers, and More Numbers

next chapter Limits, Continuity, and Elementary Functions

We’ll talk about decimal expansions of real numbers in Section 3.8 and $\pi $ in Chapter 4.

“One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second may never exceed the magnitude it approaches.” Jean d’Alembert (1717–1783). The article on “Limite” in the Encyclopédie 1754.

Note that we are not claiming that $1/(n+1) < \varepsilon $ is true for every n. We are just writing down the statement $1/(n+1) < \varepsilon $ and statements equivalent to it. The point is to discover an equivalent statement of the form $n>$ some real number. This real number is an N making (3.1) true.

How did we know that the limit is 2? The trick to finding the limit of the quotient of polynomials in n of the same degree is to divide the leading coefficients. Here, the leading coefficients of $2n^2 - n$ and $n^2-9$ are 2 and 1, respectively, so the limit is $2/1 = 2$.

You might be tempted to use logarithms on (3.12) to say that $|a|^n < \varepsilon $ if and only if $n\, \log |a| < \log \varepsilon $, or $n > \log \varepsilon /\log |a|$ (noting that $\log |a| < 0$, since $|a| < 1$). However, we have not yet developed the theory of logarithms! We will define logarithms on p. 300 in Section 4.7.

It turns out that $\pm \infty $ form part of a number system called the extended real numbers, which consists of the real numbers together with the symbols $+\infty = \infty $ and $-\infty $. One can define addition, multiplication, and order in this system, with certain exceptions (such as subtraction of infinities is not allowed). If you take measure theory, you will study this system.

For each $n \in \mathbb {N}$, define $f_n :[0,\infty ) \rightarrow [0,\infty )$ by $f_n(x) = \sqrt{a_n + x}$. Then $\alpha _1 = f_1(0)$, $\alpha _2 = f_1(f_2(0))$, $\alpha _3 = f_1(f_2(f_3(0)))$, and in general, $\alpha _{n} : = (f_1 \circ f_2 \circ \cdots \circ f_n)(0)$.

“If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.” Niels Abel (1802–1829) [225]. (Of course, nowadays series are “rigorously determined”—this is the point of this section!)

If p(x) and $q(x) = (x-r_1)(x-r_2) \cdots (x-r_n)$ are polynomials in which the $r_i$ are distinct constants and the degree of p is less than n, the method of partial fractions supposes that

$$ \frac{p(x)}{q(x)} = \frac{c_1}{x-r_1} + \frac{c_2}{x-r_2} + \cdots + \frac{c_n}{x-r_n} $$

and then solves for the constants $c_1,\cdots , c_n$. If there is a repeated factor, say $(x - r_1)^2$ in q(x), then the term $c_1'/(x - r_1)^2$ is added. It would be advantageous to review the method of partial fractions from a calculus book. In the above example, $p(x) = 1$ and $q(x) = x(x + 1)$.

“The sum of an infinite series whose final term vanishes perhaps is infinite, perhaps finite.” Jacob Bernoulli (1654–1705) Ars conjectandi.

This series is usually handled in elementary calculus courses using the technologically advanced (mathematically speaking) integral test, but Cauchy’s condensation test gives one way to handle such series without knowing any calculus!

Hui Lin was a student in my fall 2014 real analysis course, and he discovered this very interesting test. I thank him for allowing me to present his work.

There are many other nonexamples, such as $a_k(n) = 1/(n + k)$.

“To what heights would science now be raised if Archimedes had made that discovery! [${ =}$ the decimal system of numeration or its equivalent (with some base other than 10)].” Carl Gauss (1777–1855).

In the notation from Problem 5 on p. 145, $2^\mathbb {N}$ would be denoted by $\{0,1\}^\mathbb {N}$, so “2” is shorthand for the two-element set $\{0,1\}$.

Title: Infinite Sequences of Real and Complex Numbers
Author: Paul Loya
Publisher: Springer New York
Book: Amazing and Aesthetic Aspects of Analysis
Print ISBN: 978-1-4939-6793-3

Electronic ISBN: 978-1-4939-6795-7

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-1-4939-6795-7_3

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