Sequences and Series

I.   Midterm II

II.  Sequences

Exercises:  Find the next term and describe the pattern:

A)  2, 4, 6, 8, 10, ...

B)  1, 4, 9, 16, 25, ...

C)  3, 7, 15, 31, 63, ...

D)  1, -1/2, 1/6, -1/24, 1/120, -1/720, ...

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.

Consider the function f(n) = 2n + 1.  this function describes the sequence

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

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

III.  Finding the General Element of a Sequence 

One technique for finding the general element a_n is to list the number 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.

Exercise:  Find the general element a_n in the exercises listed above.

IV.  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 a_n = a_{n - 1} + a_{n - 2}

This is called the Fibonacci sequence and the terms are

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

V.  Sigma Notation and Series

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

s_n =  a₁ + a₂ + a₃ + ... + a_n

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:  e^x = 1 + x + x²/2!   x³/3!  ... xⁿ/n! + ... =