4/28/2009
1
Infinite Sequences and
Summation Notation
MATH 101
College Algebra
S. Rook
2
Overview
• Section 8.1 in the textbook:
– Infinite Sequences
– Factorials
– Partial Sums & Summation Notation
Infinite Sequences
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Community
Ask the community
Ask the community for help and clear up your study doubts
University Rankings
Discover the best universities in your country according to Docsity users
Free resources
Our save-the-student-ebooks!
Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors
Infinite Sequences, Factorials, and Summation Notation in College Algebra, Study notes of Algebra
This document from a college algebra course covers the concepts of infinite sequences, factorials, and summation notation. It includes definitions, examples, and exercises for calculating terms of infinite sequences, understanding factorial notation, and evaluating partial sums using summation notation.
Typology: Study notes
Pre 2010
1 / 6
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Infinite Sequences, Factorials, and Summation Notation in College Algebra and more Study notes Algebra in PDF only on Docsity!
Infinite Sequences and
Summation Notation
MATH 101
College Algebra
S. Rook
2
Overview
- Section 8.1 in the textbook:
- Infinite Sequences
- Factorials
- Partial Sums & Summation Notation
Infinite Sequences
4
Infinite Sequences
- We have discussed finite ( countable ) lists of numbers when constructing a table of values: - Given a function f( x ), pick values of x to get f( x ) - We do this about 2 or 3 times to get an idea what f( x ) looks like - Represents only a subset of the values of f( x )
- Infinite Sequence: a function whose domain is the natural numbers. The results that are generated from a sequence are its terms
- There are many infinite sequences of interest to mathematicians and scientists - Prime numbers, Fibonacci numbers, etc.
5
Terms of a Sequence
- The nth^ term of a sequence also called
the general term is usually written
an = f(n)
- Given a natural number k such that
1 ≤ k ≤ n, we can find the k th^ term of the
sequence by simply substituting a k = f( k )
6
Alternating Sequences
- Alternating Sequence: a sequence in
which subsequent terms change from
positive to negative or vice versa
- Has the general term an = (-1)n + 1^ · f(n)
- Substitute as before to evaluate a term
Factorials
11
Factorials
- Suppose we were give the recursive
sequence an = n · an – 1; a 1 = 1
n = 2: a 2 = 2 · a 1 = 2 · 1 = 2 n = 3: a 3 = 3 · a 2 = 3 · (2 · 1) = 3 · 2 = 6 n = 4: a 4 = 4 · a 3 = 4 · (3 · 2 · 1) = 4 · 6 = 24 : : an = n · (n – 1) · (n – 2) · … · 2 · 1
12
Factorials (Continued)
- an = n · (n – 1) · (n – 2) · … · 2 · 1 is used often enough that it is given the special name factorial and written as n! n! means the product of n down to 1 3! = 3 · 2! = 3 · 2 · 1! = 3 · 2 · 1 = 6 n! = n · ( n – 1)! 1! AND 0! are both equivalent to 1
- We can use factorials when performing Algebraic operations
Factorials (Example)
Ex 3: Evaluate by hand:
a) c)
b)
13
Partial Sums and Summation
Notation
15
Partial Sums
- We have seen how to generate successive terms from the sequence an = f(n)
- Another important series concept is the summation of these terms
- The summation of the nth^ term is called the nth partial sum denoted Sn S 1 = a 1 S 2 = a 2 + a 1 S 3 = a 3 + a 2 + a 1 : Sn = an + an-1 + … + a 2 + a 1 - Sn is called the sequence of partial sums or a series