8.1 Sequences And Series
A sequence is a functions whose domain is the set of positive integers, sequences are usually written using subscript notation.
An infinite sequence is a function whose domain is the set of positive integers. The function values
a₁, a₂, a₃ a₄, ..., an, ...
are the terms of the sequence. If the domain of a function consists of the first n positive integers only, then the sequence is a finite sequence.
Sometimes it is convenient to have a sequence begin with the 0th term, having the domain a set of non-negative integers.
An infinite sequence is a function whose domain is the set of positive integers. The function values
a₁, a₂, a₃ a₄, ..., an, ...
are the terms of the sequence. If the domain of a function consists of the first n positive integers only, then the sequence is a finite sequence.
Sometimes it is convenient to have a sequence begin with the 0th term, having the domain a set of non-negative integers.
Writing the Terms of a Sequence
Simply listing the first few terms is not sufficient to define a unique sequence- the nth term must be given.
1/2, 1/4, 1/8/ 1/16, ... 1/2ⁿ, ...
1/2, 1/4, 1/8, 1/15, ..., 6/(n + 1)(n² - n + 6), ...
1/2, 1/4, 1/8/ 1/16, ... 1/2ⁿ, ...
1/2, 1/4, 1/8, 1/15, ..., 6/(n + 1)(n² - n + 6), ...
Finding the nth Term of a Sequence
Some sequences are defined recursively. To define a sequence recursively, one or more of the first few terms must be given. All other terms are then defined using previous terms.
A Recursive Sequence
Something Similar to the Fibonacci Sequence
Factorial Notation
Some sequences involve terms defined with special types of products called factorials.
If n is a positive integer, then n factorial is defined as
n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n - 1) ⋅ n.
Zero factorial is defined as 0! = 1
Some sequences involve terms defined with special types of products called factorials.
If n is a positive integer, then n factorial is defined as
n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n - 1) ⋅ n.
Zero factorial is defined as 0! = 1
Writing the Terms of a Sequence Involving Factorials
Simplifying Factorial Expressions
Summation Notation
There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase letter sigma, written as ∑.
The sum of the first n terms of a sequence is represented by
n
∑ai = a1 + a2 + a3 + a4 + ⋅⋅⋅ + an
i = 1
where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.
There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma notation because it involves the use of the uppercase letter sigma, written as ∑.
The sum of the first n terms of a sequence is represented by
n
∑ai = a1 + a2 + a3 + a4 + ⋅⋅⋅ + an
i = 1
where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation.
Sigma Notation for Sums
Series
Many applications involve the sum of the terms of a finite or an infinite sequence. Such a sum is called a series.
Consider the infinite sequence a1, a2, a3, ... ai, ...
The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by.
n
a1 + a2 + a3 + ⋅⋅⋅ + an = ∑ai
i = 1
The sum of all the terms of the infinite sequence is called an infinite series and is denoted by
∞
a1 + a2 + a3 + ⋅⋅⋅ + an = ∑ai
i = 1
Many applications involve the sum of the terms of a finite or an infinite sequence. Such a sum is called a series.
Consider the infinite sequence a1, a2, a3, ... ai, ...
The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by.
n
a1 + a2 + a3 + ⋅⋅⋅ + an = ∑ai
i = 1
The sum of all the terms of the infinite sequence is called an infinite series and is denoted by
∞
a1 + a2 + a3 + ⋅⋅⋅ + an = ∑ai
i = 1
Finding the Sum of a Series