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Notes for Math 220, Calculus 2

Brenton LeMesurier

Section 5.1 Sequences

References.

An infinite sequence or just sequence is an infinite list of of numbers with an order of first, second, third, etc.:
\begin{equation*} \{ a_1, a_2, a_3, \dots , a_n, \dots \} \end{equation*}
For example,
\begin{equation*} \{ 1, 1/2, 1/3, 1/4, \dots , 1/n, \dots \} \end{equation*}
The subscript (here \(n\)) is the index.
A sequence can also be thought of as a function \(a(n)\) whose domain is the natural numbers, so that the index \(n\) is the argument of this function.
The individual numbers in a sequence are called the terms of the sequence, and \(a_n\) is sometimes called the \(n\)-th term.
One compact notation for the above is \(\{a_n\}_{n=1}^\infty\text{,}\) and sometimes a sequence is specified by just a formula for the general term \(a_n\text{.}\) For example, the above sequence can be described by \(a_n=1/n\text{.}\)

Other starting points.

The index can also start at any integer \(n_0\text{,}\) and \(0\) is another common starting point:
\begin{equation*} \{a_n\}_{n=0}^\infty = \{ a_0, a_1, a_2, a_3, \dots , a_n, \dots \} \end{equation*}
Since starting at \(n=1\) is the most common case, this is also denoted as just \(\{a_n\}\text{,}\) but for any other case we must be explicit, using \(\{a_n\}_{n=n_0}^\infty\) to denote a sequence that starts with term \(a_{n_0}\text{.}\)
For example the sequence
\begin{equation*} \left\{ 1, \frac12, \frac14, \frac18, \dots \right\} \end{equation*}
is most elegantly described as \(\{1/2^n\}_{n=0}^\infty\text{.}\) To specify this with a formula for the general term \(a_n\text{,}\) we now must specify the range of index values:
\begin{equation*} a_n = 1/2^n, \quad n \geq 0. \end{equation*}

Further examples.

  • The natural numbers themselves are a sequence:
    \begin{equation*} \{a_n\} = \{n\} = \{n\}_{n=1}^\infty = \{1, 2, 3, \dots \} \end{equation*}
  • The sequence with terms given by \(a_n = \frac{n}{n+1}\) can be written in the forms
    \begin{equation*} \left\{ \frac{n}{n+1} \right\} = \left\{ \frac{n}{n+1} \right\}_{n=1}^\infty = \left\{ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \dots \frac{n}{n+1}, \dots \right\} \end{equation*}
  • The sequence of the digits of \(\pi\) starts out as \(\{ 3, 1, 4, 1, 5, 9, 2, 6, 5, \dots \}\text{.}\) This time there is not a simple direct formula for \(a_n\text{,}\) but there is a way of specifying how to compute the digits.
Find an expression for the general term of a sequence that starts
\begin{equation*} \left \{ {2}/{5}, - {4}/{7}, {8}/{9}, - {16}/{11},\dots \right\} \end{equation*}
It helps to consider three sequences that are “ingredients” of the above one: \(\{ 2, 4, 8, 16,\dots \}\text{,}\) \(\{ 5, 7, 9, 11, \dots \}\text{,}\) and \(\{ 1, -1, 1, -1,\dots \}\)

Further examples: Recursion.

Another case where there is not a direct formula for \(a_n\text{,}\) but there is a procedure for computing the terms in order, is the sequence with
\begin{gather*} a_1=1, a_2=\frac{a_1+ {2}/{a_1}}{2}=\frac{1+ {2}/{1}}{2}=1.5,\\ a_3=\frac{a_2+ {2}/{a_2}}{2}=\frac{1.5+{2}/{1.5}}{2}=1.41\bar{6}, \dots, a_{n+1}=\frac{a_n+{2}/{a_n}}{2}, \dots \end{gather*}
Some more terms of this sequence, to ten decimal places, are
\begin{equation*} a_4=1.4142156863, a_5=1.4142135624, a_6=1.4142135624, \dots \end{equation*}
The values seem to be approaching \(\sqrt{2}=1.414213562373\dots\text{.}\)
This is indeed the case, and the above calculation is an example of an efficient way to compute square roots: \(a_6\) is already accurate to 10 decimal places!
This is an example of a recursively defined sequence, a central concept in computer science and methods of accurate calculation of the numerical values of functions.

More recursion: The Fibonacci sequence.

Probably the most famous example of a recursively defined sequence is the one devised by Fibonacci as a model of population growth (before calculus or exponential functions were known):
\begin{equation*} f_1=1, f_2=1, f_3 = f_2 + f_1 = 2, f_4 = f_3 + f_2 = 3, \dots, f_n = f_{n-1} + f_{n-2}, \dots \end{equation*}
The final, general formula only works for \(n \geq 3\text{,}\) once the process has been initiated with the first two values.
We can compactly describe this as
\begin{equation*} f_1=1, f_2=1, f_n = f_{n-1} + f_{n-2} \text{ for } n \geq 3. \end{equation*}

The limit of a sequence.

Sequences often arise as successively better approximations of some value, like the above recursive sequence with terms approaching \(\sqrt{2}\text{.}\) If the terms of a sequence \(\{a_n\}\) approach a value \(L\) as \(n\) increases, this value is the limit of the sequence:

Definition 5.1.2. The Limit of a Sequence.

If for every positive number \(\epsilon\text{,}\) there is an integer \(N\) so that whenever \(n > N\text{,}\) we have \(|a_n-L| < \epsilon\) then we say that the sequence \(\{a_n\}\) has limit \(L\), denoted
\begin{equation*} \lim_{n \to \infty} a_n = L, \quad \text{or} \quad a_n \to L \; \text{as} \; n \to \infty. \end{equation*}
Compare to the precise definition of \(\ds\lim_{x \to \infty} f(x)\text{,}\) as seen in Section 4.6 of OpenStax Calculus Volume 1 2 : the similarities will allow us to adapt many methods for computing the limit of a function at infinity (as with horizontal asymptotes) to computing the limits of sequences.

Infinite Limit of a Sequence.

As with functions, sequences can also have infinite limits:

Definition 5.1.3.

If for every number \(M\) there is an integer \(N\) so that whenever \(n>N\text{,}\) we have \(a_n>M\) then we say that the sequence \(\{a_n\}\) has limit infinity, denoted
\begin{equation*} \lim_{n \to \infty} a_n = \infty, \qquad \text{or} \qquad a_n \to \infty \; \text{as} \; n \to \infty. \end{equation*}
A limit of \(-\infty\) is similarly defined.
Note that sequences can have no limit at all, not even an infinite one.
\(\{ (-1)^n \}_{n=0}^\infty = \{ 1, -1, 1, -1, \dots \}\)

Sequence limits using limits at infinity of functions.

Often the limit of a sequence is the same as the limit at infinity of a function, and can be computed as we compute values for horizontal asymptotes.
The above fact allows many methods for limits of functions to be used with sequences, including l'Hôpital's Rule (see Section 4.8 of OpenStax Calculus Volume 1 3 ). In fact, limits of sequences, along with improper integrals, are two of the most important uses of that Rule.
For \(\ds a_n = \frac{\ln n}{n}\text{,}\) we can use \(\ds f(x) = \frac{\ln x}{x}\text{,}\) and
\begin{equation*} \lim_{n \to \infty} \frac{\ln n}{n} = \lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{(\ln x)'}{(x)'} = \lim_{x \to \infty} \frac{1/x}{1} = 0. \end{equation*}
So strangely, it is effectively possible to differentiate in the variable \(n\text{,}\) though its values are originally only integers!

Limit laws: much as for functions.

If \(\{a_n\}\) and \(\{b_n\}\) are convergent sequences with \(a_n \to A\text{,}\) \(b_n \to B\text{,}\) and \(c\) is a constant, then
\begin{align*} a_n + b_n \amp \to A+B\\ a_n - b_n \amp \to A-B\\ a_n \cdot b_n \amp \to A \cdot B\\ c \cdot a_n \amp \to c \cdot A\\ \frac{a_n}{b_n} \amp \to \frac{A}{B}, \quad \text{ if } B \neq 0\\ a_n^p \amp \to A^p, \quad \text{ if } p > 0 \text{ and all } a_n > 0 \end{align*}
And we again get a Squeeze Theorem:
This can be used to show that
\(\ds \frac{(-1)^n}{n} \to 0.\)
For what values of \(r\) is the geometric sequence \(\{r^n\}\) convergent, and what are the limits?
Look for both finite and infinite limits.

Definition 5.1.11. Increasing, Decreasing and Monotonic sequences.

  • Sequence \(\{a_n\}\) is increasing [a.k.a. non-decreasing] if for every index value \(n\text{,}\) \(a_n \leq a_{n+1}\text{.}\)
    Note that successive values do not have to be larger than the previous one; just no smaller; if further \(a_n < a_{n+1}\) and we want to note that, the sequence is called strictly increasing.
  • Sequence \(\{a_n\}\) is decreasing [a.k.a. non-increasing] if for every index value \(n\text{,}\) \(a_n \geq a_{n+1}\text{.}\)
  • A sequence is monotonic if it is either increasing or decreasing.
For the last two cases, there are also “strictly” versions.
An important type of increasing sequence is those produced by successively adding positive values, like these recursively defined sequences:
  • \(s_0=1\text{,}\) \(s_n = s_{n-1} + 1/n!\) for \(n \geq 1\) so \(s_n = 1/0! + 1/1! + 1/2! + \cdots + 1/n!\text{,}\) and
  • \(s_0=0\text{,}\) \(s_n = s_{n-1} + 1/2^n\) for \(n \geq 1\) so \(s_n = 1/2 + 1/4 + 1/8 + \cdots + 1/2^n\text{.}\)

Definition 5.1.12. Sequences bounded above, or below, or both.

  • Sequence \(\{a_n\}\) is bounded above [by \(M\)] if \(a_n \leq M\) for all index values \(n\text{.}\)
  • Sequence \(\{a_n\}\) is bounded below [by \(M\)] if \(a_n \geq M\) for all index values \(n\text{.}\)
  • A sequence is bounded if it is both bounded above and bounded below.
It can be shown that each of the two increasing sequences above is also bounded above; even more obviously, each is bounded below, by zero. This helps to establish that each has a limit, because of: This shows that each of the two previous examples has a limit; we will find the values of those limits later in this chapter.

Study Guide.

Study Calculus Volume 2, Section 5.1 4 ; in particular
  • The definition of an Infinite Sequence
  • The definitions of the Limit of an Infinite Sequence (a lot like a horizontal asymptote!) and of Convergence/Divergence
  • The definition of Bounded, Increasing/Decreasing and Monotone sequences
  • Theorem 1, connecting limits of sequences back to limits at infinity of functions (horizontal asymptotes again!)
  • Theorem 2, which shows that limits of sequences have a lot of properties in common with limits of functions
  • Theorems 3 and 4; more properties in common with limits of functions
  • The Monotone Convergence Theorem (Theorem 6)
  • Examples 1, 2, 3, 4, 5, 6
  • Checkpoints 1, 2, 3, 4, 5, 6
  • and one or several exercises from each of the following groups: 1–4, 8–11, 13, 14&15, 27–29, 31–34.
openstax.org/books/calculus-volume-2/pages/5-1-sequences
openstax.org/books/calculus-volume-1/pages/4-6-limits-at-infinity-and-asymptotes
openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule
openstax.org/books/calculus-volume-2/pages/5-1-sequences
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