The Third Lesson on Analysis

last updated 5apr16, previously 1jul13, 5apr11, corrected 4apr12

Commentary on Limits of Sequences and Series

\begin{document} \maketitle \section{Introduction} This lesson explains how to read Chapters 14 and 15 profitably as a supplement to the online course notes. It is not necessary to completely understand both approaches to the rudimentary definitions of \textit{ limits, sequences, series}, and you use those given in the course on homework and tests. \subsequence{Convergence of monotone sequences} The textbook defines reals in term of the \textit{ Least Upper Bound Principle} (LUBP). So the least upper bound of a set is the first significant example of a limit. But this is not the best starting point, because not all least upper bounds are the limit of an \textbf{ infinite} sequence. The confusion between a sequence and an infitie sequence causes a great deal of mischief when you first study this subject rigorously. Definitions are inclusive of special cases where one would not ordinarily think there is an issue. For a finite set of reals, for example, you can order them in increasing order starting at the mininum and ending at the maximum. There is nothing about limits here. But, consider the constant sequence $(\forall n)(1=x_n)$. Customary definitions of a limit would say that $1=lim_{n\rightarrow \infty} x_n $. This is done because on many occasions one considers infinite sequences, but which may have gobs of values equal. One can never assume that all the members of a sequence are different! \subsection{Sequences versus Series } The definitions and theorems are given in terms of sequences, but series (and continuous functions) is where they are most commonly found. So we'll bridge this disconnect by giving examples also in terms of series and functions. The Calculus II prerequisite for this course is now in effect. You already know how to \textit{calculate} limits of sequences and series, but now you learn how to \textit{analyse} the reasons behind your knowledge. \subsubsection{Definition of a Sequence} A sequence if a function $f:\mathbb{N}\rightarrow \mathbb{R}$ but written \[ \{x_n\}_{n=1}^\infty \mbox{ where } x_n = f(n) \] We drop parts or all of the decorations and sometimes just write "the sequence $x_n$ ". The textbooks notations $ \lt x \gt $ is non-standard and hard to do in MathML, and TeX, because of the angle brackets. We therefore will not use the textbook notation in this course. \subsubsection{Definition of a Limit of a Sequence} We give two versions of the same definition. \begin{eqnarray*} x_n &\mbox{ has a limit if } & (\exists a)(\forall \epsilon > 0)(\exists N > 0)(\forall n > N) (|x_n - a|<\epsilon)\\ a = lim_{n\rightarrow \infty} x_n &\mbox{ if } & (\forall \epsilon > 0)(\exists N > 0)(\forall n)(n > N \Rightarrow |x_n - a|<\epsilon)\\ \end{eqnarray*}
Question 1.
Write our symbolically what it means for a sequence $x_n$ \textbf{ not } to have a limit.
Question 2.
Write out in words what your answer to Question 1 says. \subsubsection{Definition of a Series} Given a sequence $\{x_n\}$, its series, $\sum_{n=1}^\infty x_n$, is numerically equal the limit of its partial sums, if this sequence has a limit. \[ lim_{n\rightarrow \infty} \sum_{i=1}^n x_i \] We also say that a series \textit{ converges } if this limit exists. When the limit does not exist, the series just consists of the sequence of \textit{ partial sums} \[ s_n = \sum_{i=1}^n x_i \] which happens not to have a limit. So, a series is itself a sequence derived from another sequence. I might have subseries that do converge. The trivial example for this is the series $1-1+1-1+1-1 ....$.
Question 3.
Consider the series $\sum (-1)^n$. Why does it not converge?
Question 4.
Consider the series $\sum (-1)^n$. Which subseries converge? Added since 5apr16. \subsection{Vernacular Language for Dealing with Limits} The customary informal way of wording $lim_{x\rightarrow \infty} x_n = a$ as something like "The terms of the sequence get arbitrarily close to $a$" is too vague to be much use. That is, unless you're willing to make \textit{arbitarily} take on the technical meaning above. But that is counterproductive to the purpose of using the vernacular in the first place. Another way is to parse pieces of the definition as follows. The $\forall \epsilon > 0$ could be translate into "up to any given decimal place". The phrase $(\exists N)(\forall n > N)$ can be shortened by using the notion of a \textit{tail of a sequence}. Thus, thinking of $a$ expanded into its decimal form we would say, in a somewhat legalistic manner, that \textbf{Vernacular Definition"} \textit{ Up to any given decimal there is a tail of the sequence, every term of which agrees with $a$ up the that decimal place.} \textbf{Exercise} Formulate a similarly accurate way saying that a sequence does not have a limit in the vernacular.(Do not use any symbolic expressions.) \textbf{Exercise} Use a vernacular argument a lawyer would agree to that the sum of two convergent sequences converges to the sum of their limits. \section{Null Sequences} The arithmetic of convergent, divergent, and non-convergent (all different!) sequences is much simplified by a small set of lemmas on \textit{null sequences}. Unfortunately, this field does not have a unified treatment, and so in a future course, you may not be allowed to used the properties of null sequences to prove theorems. \subsection{Definition of Null Sequence} A null sequence is, in effect, an infinite sequence which converges to 0. Thus the sequence $ (\forall n) x_n = 0 $ is not infinite, though it trivially converges to 0. We could also demand that all the terms of a null sequence are different, but that would be inconvenient in some cases. Of course, a sequence whose set of values is infinite, must have an subsequence without repetitions. Let's just not forget this subtlety, but we shouldn't interrupt a thought by saying each time. We therefor define a \textit{null sequence} to be an infinite sequence to be null if \[ (\forall \epsilon > 0)(\exists N )(\forall n > N) (|\theta_n| < \epsilon )\] And $x_n$ is \textbf{not} a null sequence if, first of all \[ (\exists \epsilon > 0)(\forall N)(\exists n > N) (|x_n| \ge \epsilon) \], which is better expressed by \[ (\exists \epsilon > 0)(\exists \mbox{ infinite subsequence } x'_n)(|x'_n| \ge \epsilon) \] and which is what we mean by the common speech expression: \textit{ infinitely many terms of the sequence stay a fixed distance away from 0}. \textbf{Exercise} Use the vernacular described above to say what it means to be a null sequence, and not to be a null sequence in term of decimal places. \subsection{Arithmetic of Null Sequences} Recall the rules of zero, such as $0 \pm 0 = 0$ and $0 \times 0 = 0$. But $\frac{0}{0}$ stays undefined because there is no definition that's consistent. There is also $a \pm 0 =a$ and $a \times 0 = 0$. These all translate into a similar set of lemmas concerning null functions, which can be expressed memorably if we agree that by writing $\theta_n$ we mean a null sequence. \textbf{Lemma} If $a_n$ is a bounded sequence, and $\theta_n$ is a null sequence the $\theta'_n = a_n \times \theta_n $ is a null sequence as well. \textbf{Exercise.} Can you prove this? Hint: Given an $\epsilon$ for $\theta'_n$, what $\epsilon$ should you choose for $\theta_n$ to "work"? \subsection{Using Null Sequences for Limits} In the vernacular, we say that $\lim_{n\rightarrow \infty} x_n = a$ we say that "$x_n$ gets arbitrarily close to $a$ for $n$ sufficiently large". This now can be said with full accuracy in terms of a null sequence \textbf{Definition} $\lim_{n\rightarrow \infty} x_n = a$ means that $x_n = a + \theta_n$ for some null sequence. \textbf{Lemma} Why is this definition equivalent to the original definition we gave above? \textbf{Example:} \begin{eqnarray*} \mbox{ If } x_n &=& a + \theta'_n \\ \mbox{ and } y_n &=& b + \theta"_n \\ \mbox{ then } x_n \pm y_n &=& (a \pm b) + (\theta'_n \pm \theta"_n) \\ \therefore x_n \pm y_n &=& a \pm b + \theta'''_n \\ \end{eqnarray*} yields more intuitive proof about the sum and difference of convergent sequences because it emphasizes that the sequence is an approximation of its limit. \textbf{Exercise} Formulate a lemma concerning null sequences which makes the quotient rule for limits easy to prove. \[ \lim x_n = a \wedge \lim y_n = b \ne 0 \Rightarrow \lim \frac{x_n}{y_n}=\frac{a}{b} \]. \subsection{Diverging Sequences} A sequence can fail to have a limit in many ways. One useful way is to say that it \textit{diverges} to $\infty$ (or to $-\infty$) as follows. \textbf{Definition} A positive sequence $x_n$ diverges (to $\infty$) means that \[ (\forall K>0)(\exists N)(\forall n > N)(x_n > K) \]. \textbf{Exercise} Show that a positive sequence $\theta_n$ is a null sequence if and only if $\frac{1}{\theta_n}$ diverges to $\infty$. How's that for solving the riddle of why $\infty = \frac{1}{0}$? \textbf{Exercise} What does it mean that a positive sequence neither converges nor diverges? What does it do, especially if it is also infinite. Can you give some examples of such nasty customers. \end{document}