1.2 Sequences in General – Series

1. Sequences

A sequence is an ordered list of numbers. The numbers in the list are called terms, and their order matters.

For example, consider the sequence

$$2,\quad 5,\quad 13,\quad 5,\quad -4,\quad \dots$$

Term $2$ $5$ $13$ $5$ $-4$
Position 1st 2nd 3rd 4th 5th

Usually, the terms of a sequence follow a recognizable pattern. For example:

  • $0,2,4,6,8,10,\dots$ — even numbers
  • $1,3,5,7,9,11,\dots$ — odd numbers
  • $5,10,15,20,25,\dots$ — positive multiples of $5$
  • $2,4,8,16,32,\dots$ — powers of $2$

We use the notation $u_n$ to describe the $n$-th term of a sequence. Thus, the terms of a sequence are written as

$$u_1,\quad u_2,\quad u_3,\quad u_4,\quad u_5,\quad \dots$$

For example, $u_1$ is the first term, $u_2$ is the second term, and $u_{10}$ is the tenth term.

2. Series and Partial Sums

A series is a sum of terms of a sequence.

The sum of the first $n$ terms is denoted by $S_n$:

$$S_n=u_1+u_2+u_3+\dots+u_n.$$

If we continue indefinitely, we write

$$S_\infty=u_1+u_2+u_3+\dots$$

The finite sums $S_1,S_2,S_3,\dots$ are called partial sums. The symbol $S_\infty$ is used for an infinite series when such an infinite sum is meaningful.

EXAMPLE 1

Consider the sequence of odd numbers

$$1,3,5,7,9,11,\dots$$

Some terms are

$$u_1=1,\qquad u_2=3,\qquad u_3=5,\qquad u_6=11,\qquad u_{10}=19.$$

The first few partial sums are

$$\begin{aligned} S_1&=1,\\ S_2&=1+3=4,\\ S_3&=1+3+5=9,\\ S_4&=1+3+5+7=16. \end{aligned}$$

For the infinite series

$$1+3+5+7+\dots,$$

the partial sums grow without bound, so in this case we write $S_\infty=+\infty$.

3. Sigma Notation

Instead of writing a long sum such as

$$u_1+u_2+u_3+u_4+u_5+u_6+u_7+u_8+u_9,$$

we may write

$$\sum_{n=1}^{9}u_n.$$

This means: add all the terms $u_n$ as $n$ runs from $1$ to $9$.

More generally,

$$\sum_{n=1}^{k}u_n=u_1+u_2+\dots+u_k.$$

The lower limit does not have to be $1$. For example,

$$\sum_{n=4}^{9}u_n=u_4+u_5+u_6+u_7+u_8+u_9.$$

EXAMPLE 2

(a)

$$\sum_{n=1}^{3}2^n =2^1+2^2+2^3 =2+4+8 =14.$$

(b)

$$\sum_{n=1}^{4}\dfrac{1}{n} =1+\dfrac12+\dfrac13+\dfrac14 =\dfrac{25}{12}.$$

(c)

$$\sum_{k=1}^{3}\dfrac{1}{2^k} =\dfrac12+\dfrac14+\dfrac18 =\dfrac78.$$

(d)

$$\sum_{n=3}^{6}(2n+1) =7+9+11+13 =40.$$

(e)

$$\sum_{x=3}^{20}\dfrac{x}{x+2} =\dfrac35+\dfrac46+\dfrac57+\dots+\dfrac{20}{22}.$$

For a long finite sum, sigma notation is usually much more convenient than writing every term.

Infinite Sum Example

An infinite series can also be written using sigma notation:

$$\sum_{n=1}^{\infty}\dfrac{1}{2^n} =\dfrac12+\dfrac14+\dfrac18+\dfrac1{16}+\dots$$

The sum never terminates, but its partial sums get closer and closer to $1$. Therefore,

$$\sum_{n=1}^{\infty}\dfrac{1}{2^n}=1.$$

We will study the conditions under which an infinite geometric series has a finite sum in Section 1.4.

4. Two Ways to Describe a Sequence

There are two basic ways to describe a sequence:

  1. by a general formula for $u_n$;
  2. by a recursive relation.

A. General Formula

A general formula describes $u_n$ directly in terms of $n$.

For example:

$$u_n=2n.$$

Then

$$u_1=2,\qquad u_2=4,\qquad u_3=6,\qquad \dots$$

so the sequence is

$$2,4,6,8,10,\dots$$

EXAMPLE 3

If

$$u_n=n^2,$$

then the sequence is

$$1,4,9,16,25,\dots$$

If

$$u_n=2^n,$$

then the sequence is

$$2,4,8,16,32,\dots$$

B. Recursive Relation

A recursive definition gives an initial term, such as $u_1$, and then defines each new term using one or more preceding terms.

For example:

$$u_1=10,\qquad u_{n+1}=u_n+2.$$

This says that the first term is $10$, and each subsequent term is obtained by adding $2$ to the preceding term:

$$\begin{aligned} u_2&=u_1+2=10+2=12,\\ u_3&=u_2+2=12+2=14,\\ u_4&=u_3+2=14+2=16. \end{aligned}$$

Therefore the sequence is

$$10,12,14,16,18,\dots$$

In simple terms: start with $10$ and repeatedly add $2$ to obtain the next term.

EXAMPLE 4

Consider the recursive sequence

$$u_1=3,\qquad u_{n+1}=2u_n+5.$$

Then

$$\begin{aligned} u_2&=2(3)+5=11,\\ u_3&=2(11)+5=27,\\ u_4&=2(27)+5=59. \end{aligned}$$

Thus the sequence is

$$3,11,27,59,\dots$$

EXAMPLE 5 — Fibonacci Sequence

Sometimes a recursive relation uses the previous two terms. The most famous example is the Fibonacci sequence:

$$u_1=1,\qquad u_2=1,\qquad u_{n+1}=u_n+u_{n-1}\quad(n\geq2).$$

We add the two preceding terms to obtain the next term:

$$\begin{aligned} u_3&=u_2+u_1=1+1=2,\\ u_4&=u_3+u_2=2+1=3,\\ u_5&=u_4+u_3=3+2=5,\\ u_6&=u_5+u_4=5+3=8. \end{aligned}$$

Hence the Fibonacci sequence begins

$$1,1,2,3,5,8,13,21,34,55,\dots$$