Sequences and Series

Sequences
Example: Find the next term and describe the pattern:

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

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

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

1, 1/2, 1/6, 1/24, 1/120, 1/720, ...
Solution:

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

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

The next term is 127. These numbers are all one less than a power of two.
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 a_{n} to describe a sequence instead
of the notation f(n).

Finding the General Element of a Sequence
One technique for finding the general element a_{n} 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 a_{n} in the exercises
listed above.
Solution:

a_{n} = 2n

a_{n} = n^{2}

a_{n} = 2^{n+1}  1

a_{n} = (1)^{n+1} /n!

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
a_{1}
=1, a_{2} = 1
and
a_{n} = a_{n  1}
+ a_{n  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,...

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 n^{th}
partial sum as
s_{n} = a_{1} + a_{2} + a_{3} + ...
+ 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 +
x^{2}/2! x^{3}/3! ...
x^{n}/n! + ... =
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