Sequences and Series

1. Sequences

Example:  Find the next term and describe the pattern:

1. 2, 4, 6, 8, 10, ...

2. 1, 4, 9, 16, 25, ...

3. 3, 7, 15, 31, 63, ...

4. 1, -1/2, 1/6, -1/24, 1/120, -1/720, ...

Solution:

1. We see that the next term is 12.  We can get to the next term by adding two.

2. The next term is 36.  The terms are all squares.

3. The next term is 127.  These numbers are all one less than a power of two.

4. The next term is 1/5040.  The numbers alternate sign and the denominators are all factorials.

 Definition   A sequence is a list of numbers.  In technical terms, a sequence is a function whose domain is the set of natural numbers and whose range is a subset of the real numbers.

Example:

Consider the function

f(n) = 2n + 1

This function describes the sequence

3,5,7,9,11,...

We will usually use the notation an to describe a sequence instead of the notation f(n).

2. Finding the General Element of a Sequence

One technique for finding the general element an is to list the numbers 1,2,3,4,5,6... above the sequence and decide what do we have to do to the number 5 for example to get the fifth term.  then generalize.

Example

Find the general element an in the exercises listed above.

Solution:

1. an = 2n

2. an = n2

3. an = 2n+1 - 1

4. an = (-1)n+1 /n!

3. Recursively Defined Sequences

A recursively defined sequence is a sequence where the first term(s) are given and the next term is given in terms of the previous terms.

Example

Let

a1 =1,     a2 = 1

and

an = an - 1 + an - 2

This is called the Fibonacci sequence and the terms are

1,1,1+1 = 2,1+2 = 3,2+3 = 5,3+5 = 8,5+8 = 13,8+13 = 21,...

1,1,2,3,5,8,13,21,34,55,...

4. Sigma Notation and Series

 Definition We define a series to be the sum of the sequence.

Example:

If

1,1/2,1/4,1/8,...

is a sequence, then

1 + 1/2 + 1/4 + 1/8 + ...

is the corresponding sum.  We define the nth partial sum as

sn =  a1 + a2 + a3 + ... + an

We write this series in Sigma Notation as follows.

This is read, "The sum from 1 to infinity of 1 over 2 to the n."

Application:
ex = 1 + x + x2/2!   x3/3!  ... xn/n! + ... =