DrDelMath

SUMMARY
Chapter 8:  Sequences, Series, and Probability

Section 8.1: Sequences and Series

DEFINITION: A sequence is a function whose domain is the set of Natural Numbers.

Functional Notation : Normally if the name of a function is a and n is an element of the domain of the function, then we use a(n) to denote the unique range element which is associated with the domain element n by the function a. However, historical useage dictates a slightly different functional notation for sequenes.

If a is the name of a sequence and n is a domain element then we use a_n to denote the unique range element which is associated with the domain element n by the function a.

DEFINITION: If n is a natural number, n factorial is defined by n! = (1)(2)(3) ... (n - 1)(n).

DEFINITION: Summation Notation: The sum of the first n terms of a sequence named a is represented by
                          
                           i is the index of summation, 1 is the lower limit and n is the upper limit.

DEFINITION: The sum of the first n terms of a sequence is call the n^th partial sum of the sequence.

DEFINITION: The sum of all the terms of a sequence is called an infinite series.

Notation: The n^th partial sum of a sequence named a is symbolized by and the infinite series associated with the sequence named a is symbolized by

Section 8.2: Arithmetic Sequences and Partial Sums

DEFINITION: A sequence a is an arithmetic sequence if the difference of consecutive terms of a is the same for all terms of the sequence.
DEFINITION: The difference between consecutive terms of an arithmetic sequence is called the common difference of the sequence.
Fact: The n^th term of an arithmetic sequence named a is given by the rule a_n = dn + c where d is the common difference and c is a constant.
Observe that if we used normal functional notation the rule would be written as a(n) = dn + c which looks like the normal linear function which is normally presented as f(x) = mx + b.

Section 8.3: Geometric Sequences and Series

DEFINITION: A sequence a is a geometric sequence if the ratio of consecutive terms of a is the same for all terms of the sequence.
DEFINITION: The ratio between consecutive terms of a geometric sequence is called the common ratio of the sequence.

Fact: The n^th term of a geometric sequence named a is given by the rule a_n = a₁r^{n - 1} where r is the common ratio.
Observe that if we used normal functional notation the rule would be written as a(n) = a₁r^{n - 1} which looks like a normal exponential function which is normally presented as f(x) = ar^x.

Section 8.4: Mathematical Induction

Section 8.5: The Binomial Theorem

Section 8.6: Counting Principles

Goals -- What You Should Learn 1. How to 2. How to 3. How to 4. How to 5. How to 6. How to

Links to Supplemental Material Always Check the List of OnLine Drill and Practice 1. link 1 2. link 2 3. link 3 4. link 4 5. link 5

Section 1.7: Probability