Sequences

Calculus 2

Sequences

Discussion

One of Zeno's Paradoxes is this: a man standing in a room could never walk to a wall because he would first have to walk half the distance to the wall, then half the remaining distance, and then again half of what still remains, continuing in this way indefinitely. Describe the distances that the man walks as a sequence of numbers.

View Answer

\frac{1}{2}, \frac{1}{2}+\frac{1}{4}, \frac{1}{2}+\frac{1}{4}+\frac{1}{8}, \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16},...=\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{15}{16}, . . .

Introduction to Sequences

A sequence is a list of numbers written in a specific order. Sequences can be defined randomly, recursively, or explicitly. We index the terms of the sequence with positive integers:

a_1, a_2, a_3, \dots, a_n, \dots

A sequence can be finite or infinite — having a finite or infinite number of terms. We will be focusing specifically on infinite sequences, which we will denote by $\{a_n\}_{n=1}^\infty$ .

An explicitly defined sequence is a type of function whose inputs are term numbers and whose outputs are terms in the sequence. However, the domain of a sequence is the set of natural numbers ( $1$ , $2$ , $3$ , ...).

Some sequences have continuous function counterparts ( $a_n=3n-1$ to $f(x)=3x-1$ ) and some do not ( $a_n=(-2)^n$ or $b_n=\frac{n!}{n^2}$ ).

Infinite sequences can be represented in several ways:

$a_n = \{2, 5, 8, 11, 14, \dots \}$
$a_n = 2, 5, 8, 11, 14, \dots$
$\{a_n\} = 3n-1$
$a_n = 3n-1$

Example 1

List the first three terms in the sequence $\left\{ \frac{n}{n+1} \right\}_{n=1}^\infty.$

View Answer

First three terms: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}$

Convergence/Divergence of Sequences

Sequences, like functions, have right-end behavior that we are interested in.

If the limit of the sequence as $n$ approaches infinity exists, the sequence converges.

If the limit fails to exist, the sequence diverges.

Sometimes it is obvious whether the sequence converges or not. It is often possible to consider what a function $f(x)$ with the same definition would do as $x$ approaches infinity.

If $a_n$ has a continuous function counterpart, you can use limit techniques (including L'Hopital's Rule) to determine the limit of $f(x)$ and then apply it to the sequence $\{a_n\}$ .

Example 2

Determine whether the given sequences converge or diverge:

$a_n = -4$
$b_n = 2n+1$
$c_n = \frac{n^2+2n}{1+6n^2}$
$d_n = \cos(\pi n)$

View Answer

$a_n$ : Converges to $-4$
$b_n$ : Diverges to infinity
$c_n$ : Converges to $\frac{1}{6}$
$d_n$ : Diverges (oscillates)

Example 3

Determine whether the given sequences converge or diverge:

$a_k = \sqrt{k}$
$b_k = (-1)^k$
$c_k = k!$
$d_k = \frac{1}{(-2)^k}$

View Answer

$a_k$ : Diverges to infinity
$b_k$ : Diverges (oscillates)
$c_k$ : Diverges to infinity
$d_k$ : Converges to $0$

Growth Rates

Factorials and alternating sequences have no continuous function counterparts, so other methods must be used to determine convergence. Two useful approaches are the Squeeze Theorem and growth rates. We will focus on using growth rates.

Many sequences are fractions, and if you can determine whether the top or bottom of the fraction grows "more quickly", you can determine whether the sequence converges or diverges.

As $n \to \infty$ :

c << \ln(n) << n^p << b^n << n! << n^n \\ \text{constant} << \text{logarithm} << \text{polynomial} << \text{exponential} << \text{factorial} << \text{tetration}

Example 4

Determine whether the given sequences converge or diverge:

$a_n = \frac{n^3}{\ln(n)+1}$
$b_n = \frac{\ln(n)+n}{n^3}$
$c_n = \frac{n!}{n^3+6^n}$
$d_n = \frac{(-100)^n}{n!}$

View Answer

$a_n$ : Diverges to infinity
$b_n$ : Converges to $0$
$c_n$ : Diverges to infinity
$d_n$ : Converges to $0$

Example 5

Determine whether the given sequences converge or diverge:

$a_k = \frac{6^k+k}{5^k-k^2}$
$b_k = \frac{k!}{(2k)!}$
$c_k = \frac{\ln k}{\ln k^2}$
$d_k = \frac{7k^3+1}{k^2-4k^3}$

View Answer

$a_k$ : Diverges to infinity
$b_k$ : Converges to $0$
$c_k$ : Converges to $\frac{1}{2}$
$d_k$ : Diverges to $-\frac{7}{4}$

Sequences and Series

Alternating Series