Chapter 7: Sequences and Series

Section 7.1

Student Learning Outcomes

  • Sequences as functions whose domains are subsets of the non-negative integers
  • Common sequences defined with formulas and with recursion
  • Monotonic sequences and bounded sequences

Definition of a Sequence

A sequence of real numbers $\{a_k\}_{k=a}^\infty$ is a function from positive integers to the real numbers. For each positive integer index $k$, the output $a_k$ is called the $k$th them of the sequence.

Examples

  • $\{2+k\}_{k=1}^\infty=3, 4, 5, \dots$
  • $\{a_{k-1}+a_{k-2}\}_{k=1}^{\infty}$ and $a_1=a_2=1$, then the sequence is $1, 1, 2, 3, 5, 8, \dots$
  • $\displaystyle \left\{ \frac{(-1)^k}{k^2} \right \}_{k=1}^{\infty}= \left\{-1, \quad \frac{1}{4}, \quad -\frac{1}{9}, \quad \frac{1}{16}, \quad -\frac{1}{25}, \dots\right\}$

Recursively Defined Sequences

Some sequences are best described in terms of previous terms instead of with an explicit fomrula.

Examples

  • $\{a_{k-1}+a_{k-2}\}_{k=1}^{\infty}$ and $a_1=a_2=1$, then the sequence is $1, 1, 2, 3, 5, 8, \dots$ is a recursively defined sequence. It is called the Fibonacci sequence.

  • $k!=k(k-1)(k-2)\cdots(3)(2)(1)$ is a recursively defined function; it is called the factorial of a number.

    • For example, $3!=3\cdot2\cdot1=6$ and $5!=5\cdot4\cdot3\cdot2\cdot1$.

Recursive Definition of a Factorial

$$k!=k(k-1)!$$

Example

$$\frac{k!}{(k+1)!}$$
$$=\frac{k!}{(k+1)k!}$$
$$=\frac{1}{k+1}$$

Finding a Formula for a Sequence

Find a plausible formula for the general term of the given sequence. $$\left\{ \frac{1}{9}, \quad \frac{2}{81}, \quad \frac{1}{243}, \quad \frac{4}{6561}, \quad \frac{5}{59049}, \quad \dots \right\}$$

Notice that the top terms are almost counting by 1, $\{1, 2, 3, 4, \dots\}$, and the bottom are almost powers of 9, $\{9, 81, 729, 6561, 59049, \dots\}$.

So, the numerator would be $k$ and the denominator would be $9^k$, so the general sequence is $$\left\{\frac{k}{9^k}\right\}_{k=1}^\infty$$

Finding a Formula for a Sequence

Find a plausible formula for the general term of the given sequence. $$\left\{9, \quad -\frac{9}{5}, \quad \frac{9}{25}, \quad -\frac{9}{125}, \quad \frac{9}{625}, \quad \dots \right\}$$

This is an alternating series, meaning the sign alternates between positive and negative.

The numerator is always 9, and the bottom is multiples of 5, so...

$$\left\{(-1)^k\frac{9}{5^k}\right\}_{k=0}^{\infty} \quad \text{or} \quad \left\{(-1)^{k-1}\frac{9}{5^{k-1}}\right\}_{k=1}^{\infty}$$

Arithmetic and Geometric Sequences

An arithmetic sequence is a sequence in which each term differs by a constant additive increment.

$$\{c+dk\}_{k=0}^{\infty}=\{c, c+d, c+2d, c+3d, \dots\}$$

Examples

  • $\{5, 7, 9, 11, 13, 15, \dots\}$, $d=2$

  • $\{1, -2, -5, -8, -11, \dots\}$, $d=-3$

  • $\displaystyle \left\{1, \frac{3}{2}, 2, \frac{5}{2}, \dots \right\}$, $\displaystyle d=\frac{1}{2}$

A geometric sequence is a sequence in which each term differs by a constant multiplicative increment.

Examples

  • $\{5, 25, 125, \dots\}$, $r=5$

  • $\{1, -3, 9, -27, 81, \dots\}$, $r=-3$

  • $\displaystyle \left\{\frac{2}{3}, \frac{4}{9}, \frac{8}{27} \dots \right\}$, $\displaystyle r=\frac{2}{3}$

Bounded and Unbounded Sequences

Consider the sequence $$\left\{ \frac{\cos(\pi k)}{k} \right\}_{k=1}^{\infty}$$

The first few terms of the sequence are

$$\left\{ -1, \quad \frac{1}{2}, \quad -\frac{1}{3}, \quad \frac{1}{4}, \quad -\frac{1}{5}, \quad \cdots \right\}$$

The sequence is bounded between $\frac{1}{2}$ and $-1$.

We call $\frac{1}{2}$ the Least Upper Bound (also called the supremum, $sup$) and we call $-1$ the Greatest Lower Bound (also called the infimum, $inf$).

In [9]:
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Monotonicity of a Sequence

A sequence is called monotonic if it is either always increasing or always decreasing where:

  1. the sequence is called increasing if $a_{k+1}\geq a_k$ for all $k\geq1$;

  2. the sequence is called decreasing if $a_{k+1}\leq a_k$ for all $k\geq1$.