Sequences

Section 9.1 Sequences

Subsection 9.1.1 Zeno’s Paradox

There are three classical branches of calculus. The first two, derivatives and integrals, command the vast majority of the time and energy in most first year calculus classes. In many universities, these two topics are the entire course. However, there is a third branch of the calculus which deserves equal attention: infinite series.

In some ways, the problem of infinite series is older than the problems motivating derivatives and integrals. Infinite series go back at least to early Greek mathematics, where thinkers struggled with the puzzles of infinitiy. The most famous of those ancient puzzles is known as Zeno’s Paradox.

There are many forms of Zeno’s Paradox; I will present one relatively common version. If you wish to travel from point \(a\) to point \(b\text{,}\) then first you must travel half-way. Having gone halfway to \(b\text{,}\) you must again cover half the remaining distance. Having gone \(\frac{3}{4}\) of the way to \(b\text{,}\) there is still a distance remaining, and you still must first cover half that distance. Repeating this process gives an infinite series of halves, all of which must be traversed to travel from \(a\) to \(b\text{.}\) Since doing an infinite number of things is not humanly possible, you will never be able to reach \(b\text{.}\) Finally, since this holds for any two points \(a\) and \(b\text{,}\) movement is impossible.

Obviously, Zeno’s paradox doesn’t hold, since movement is, indeed, possible. But Zeno’s paradox has commanded the attention and imagination of philosophers and mathematicians for over 2000 years, as they struggled to deal with the infinity implicit in even the smallest movement. Infinite series is one way (though, some would argue, an incomplete way) of dealing with Zeno’s paradox.

Before defining series themselves, I need to start with infinite sequences.

Subsection 9.1.2 Definition

Definition 9.1.1.

An infinite sequence of real numbers is a set of real numbers indexed by \(\NN\text{.}\) The following are some common notations for an infinite sequence.

\begin{align*} \amp \left\{ a_n \right\}_{n \in \NN} \amp \amp \left\{ a_n \right\}_{n=1}^\infty \amp \amp \left\{ a_n \right\} \amp \amp \left\{ a_1, a_2, a_3, a_4, \ldots \right\} \end{align*}

Before moving on to examples, I want to make one more definition. The most obvious way to define a sequence is by a direct formula, such as \(a_n = n^2\text{,}\) the sequence of square numbers. The most important non-direct method is covered by the next definition.

Definition 9.1.2.

A sequence \(\{ a_n \}\) is called a recursive sequence is each element is determined by the previous element (or previous few elements). The equation that defined the recursive sequence is call a recurrence relation. Often a recurrence relation is difficult to calculate with and a direct description is desired. Such a descriptions is called the closed form description for the recursive sequence.

Let me introduce an important change in perspective for sequences. Instead of thinking of \(\NN\) as an index, I can think of a sequence \(\{a_n\}_{n=1}^\infty\) as a function.

\begin{align*} \amp f: \NN \rightarrow \RR \amp \amp f(n) = a_n \end{align*}

If I think of sequences as functions on \(\NN\text{,}\) then I can use all of the language of functions. In this way, sequences can be increasing, decreasing, monotonic, bounded above, bounded below and bounded. However, since the domain \(\NN\) is separated into discrete numbers, this function \(f\) has no continuity properties.

Even though I stated the definition for indices in \(\NN\text{,}\) I can choose another starting point. \(\{a_n\}_{n=3}^\infty\) is a sequence which starts with \(a_3\text{,}\) and \(\{a_n\}_{n=-2}^\infty\) is a sequences which starts with \(a_{-2}\text{.}\) Regardless of the starting point, sequences still always count up by 1 from the starting point.

Subsection 9.1.3 Examples of Sequences

Sequences can be entirely random, or patterned by some formula or recursion. Here are some familiar examples. I’ ll show the first few terms as well as a method of generating higher terms (either direct or recursive).

Example 9.1.3.

The sequence of natural numbers.

\begin{align*} \amp \NN = \left\{ 1,2,3,4,5, \ldots \right\} \amp \amp a_n = n \end{align*}

Example 9.1.4.

The sequence of even numbers.

\begin{align*} \amp \left\{ 2,4,6,8,10, \ldots \right\} \amp \amp a_n = 2n \end{align*}

Example 9.1.5.

The harmonic sequence.

\begin{align*} \amp \left\{ 1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5} \ldots \right\} \amp \amp a_n = \frac{1}{n} \end{align*}

Example 9.1.6.

The alternating harmonic sequence.

\begin{align*} \amp \left\{ 1,\frac{-1}{2},\frac{1}{3},\frac{-1}{4},\frac{1}{5} \ldots \right\} \amp \amp a_n = \frac{(-1)^{n+1}}{n} \end{align*}

Example 9.1.7.

The geometric sequence with common ratio \(\frac{-1}{2}\) and starting value \(1\text{.}\) (I’ve given both a direct and a recursive definition of this sequence.)

\begin{align*} \amp \left\{ 1,\frac{-1}{2},\frac{1}{4},\frac{-1}{8},\frac{1}{16} \ldots \right\} \amp \amp a_n = \left(\frac{-1}{2}\right)^n \amp \amp a_n = \left( \frac{-1}{2} \right) a_{n-1} \end{align*}

Example 9.1.8.

The arithmetic sequence with common difference 6 and starting value 1. (I’ve given both a direct and recursive definition of this sequence.)

\begin{align*} \amp \left\{ 1,7,13,19,25, \ldots \right\} \amp \amp a_n = 1 + 6n \amp \amp a_n = a_{n-1} + 6 \end{align*}

Example 9.1.9.

The Fibonacci sequence. (This is a very famous example of a recursive sequence.)

\begin{align*} \amp \left\{1, 1, 2, 3, 5, 8, 13, \ldots \right\} \amp \amp a_1 = a_2 = 1 \amp \amp a_n = a_{n-1} + a_{n-2} \end{align*}

Example 9.1.10.

The sequence of ratios of Fibonacci terms.

\begin{align*} \amp \left\{ 1,\frac{2}{1},\frac{3}{2},\frac{5}{3},\frac{8}{5} \ldots \right\} \amp \amp a_1 = 1 \amp \amp a_n = 1 + \frac{1}{a_{n-1}} \end{align*}

There are many, many sequences studied in mathematics. The Online Encyclopedia of Integer Sequences (OEIS) is a repository for interesting sequences with integer values. As of March 19, 2021, there were 342283 sequences in the OEIS.

Subsection 9.1.4 Limits of Sequences

As functions \(\NN \rightarrow \RR\text{,}\) sequences are not continuous, so I can’t ask for limits at finite values. However, since the index \(n \rightarrow \infty\text{,}\) I can ask for the long term behaviour of the sequence.

Definition 9.1.11.

The statement

\begin{equation*} \lim_{n \rightarrow \infty} a_n = L \end{equation*}

means that as \(n\) gets larger and larger without bound, \(a_n\) gets closer and closer to \(n\text{.}\) Similarly, the statement

\begin{equation*} \lim_{n \rightarrow \infty} a_n = \infty \end{equation*}

means that as \(n\) gets larger and larger without bound \(a_n\) also gets larger and larger without bound. Sequences with finite limits are convergent sequences, and all others (where the limit is either infinite or non-existant) are divergent sequences.

To understand limits, I can make great use of the perspective of sequences as functions. I know that limits of functions have many useful properties; all those properties transfer to sequences. Most importantly, for any sequences defined by a familiar functions, asymptotic analysis is still appropriate. The limits of sequences are limits at infinity, so I will use asymptotic analysis whenever possible. In addition, many other properties of sequences of functions transfer over.

Proposition 9.1.12.

If \(\{a_n\}\) and \(\{b_n\}\) are convergent sequences, then the following properties hold.

\begin{align*} \lim_{n \rightarrow \infty} (a_n + b_n) \amp = \lim_{n \rightarrow \infty} a_n + \lim_{n \rightarrow \infty} b_n \amp \amp \\ \lim_{n \rightarrow \infty} (a_n-b_n) \amp = \lim_{n \rightarrow \infty} a_n - \lim_{n \rightarrow \infty} b_n \amp \amp \\ \lim_{n \rightarrow \infty} ca_n(x) \amp = c \lim_{n \rightarrow \infty} a_n(x) \amp \amp \\ \lim_{n \rightarrow \infty} a_n =\amp \lim_{n \rightarrow \infty} a_n \lim_{n \rightarrow \infty} b_n \amp \amp \\ \lim_{n \rightarrow \infty} \frac{a_n}{b_n} \amp = \frac{\lim_{n \rightarrow \infty} a_n}{\lim_{n \rightarrow \infty} b_n} \amp \amp \\ \lim_{n \rightarrow \infty} (a_n(x))^n \amp = (\lim_{n \rightarrow \infty} a_n(x))^n \amp \amp \\ \lim_{n \rightarrow \infty} \sqrt[n]{a_n} \amp = \sqrt[n]{\lim_{n \rightarrow \infty}} a_n \amp \amp a_n \geq 0 \end{align*}

Example 9.1.13.

Here are some limits where I can simply apply the method of asymotptic analysis from limits of functions. Here are three limits which easy to argue by knowing the asymptotic order and growth of conventional functions.

\begin{align*} \lim_{n \rightarrow \infty} n^2 \amp = \infty\\ \lim_{n \rightarrow \infty} \frac{1}{n} \amp = 0\\ \lim_{n \rightarrow \infty} \frac{n+1}{n^2} \amp = 0 \end{align*}

Example 9.1.14.

Asymptotic anlysis doesn’t solve everything; some limits are still difficult to determine. One such limit is the limit definition of the number \(e\text{.}\)

\begin{equation*} \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^n = e \end{equation*}

Asymptotic analysis has trouble with this limit because the term inside the brackets approaches \(1\text{.}\) \(1\) to any power is just one, so there is an intuition that limit should be \(1\text{.}\) However, the exponent approches \(\infty\) and the term in the brackets is (at least slightly) larger then one. A very high power of a number larger than \(1\) should grow very large, so there is also an intuition that the limit should be infinite. It is not clear, by examining the limit, whether one of these two intutition is correct or whether (as is the case) the limit ends up between the two intuititon. This is another type of indeterminant form.

Example 9.1.15.

One of the example sequences was the ratio of the Fibonacci terms. If \(f_n\) are the Fibonacci terms, then the ratio is \(a_n = \frac{f_{n+1}}{f_n}\text{.}\) I can use the recursive definition of the sequence of Fibonacci terms: \(f_{n+1} = f_{n} + f_{n-1}\text{.}\) In the ratio \(a_n\text{,}\) I can replace the numerator with this recursive definition. Then some algebra lets me determine a recurivse definition for the \(a_n\text{.}\)

\begin{equation*} \frac{f_{n+1}}{f_n} = \frac{f_n}{f_n} + \frac{f_{n-1}}{f_n} \implies a_n = 1 + \frac{1}{\frac{f_n}{f_{n-1}}} \implies a_n = 1 + \frac{1}{a_{n-1}} \end{equation*}

Now I have a recursive definition for the sequences of Fibonacci ratios. Recursive definition are problematic for direct calculations of limits, but there is a useful trick for calculating these limits. Let me restate this recurvise definition.

\begin{equation*} a_n = 1 + \frac{1}{a_{n-1}} \end{equation*}

The trick is to take the limit of both sides of the equation. On the left, I have the limit that I want to calculate. On the right, I can use the limit laws to take the limit inside the various operations. Then, the limit of \(a_{n-1}\) is also the limit I want to calculate. Let me lable the result of this limit as \(\phi\text{.}\)

\begin{align*} a_n \amp = 1 + \frac{1}{a_{n-1}}\\ \lim_{n \rightarrow \infty} a_n \amp = 1 + \frac{1}{\lim_{n \rightarrow \infty} a_{n-1}}\\ \phi \amp = 1 + \frac{1}{\phi} \implies \phi^2 = \phi + 1 \implies \phi^2 - \phi - 1 = 0 \end{align*}

Now I have an equation for the value of the limit. I proceed to solve the equation.

\begin{align*} \phi \amp = 1 + \frac{1}{\phi} \implies \phi^2 = \phi + 1 \implies \phi^2 - \phi - 1 = 0\\ \phi \amp = \frac{1 \pm \sqrt{1+4}}{2} = \frac{1 \pm \sqrt{5}}{2} = \frac{1 + \sqrt{5}}{2} \end{align*}

Via the recursive definition, I’ve managed to calculate the limit. This number \(\phi\) is the celebrated golden ratio, and I’ve proved that the limit of the Fibonacci terms is the golden ratio.

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