Chapter 9: Sequences, Series, and Probability

An infinite sequence is a function whose domain is the set of positive integers. The function values

are the terms of the sequence. If the domain of the function consists of the first n positive integers only, the sequence is a finite sequence.

Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. The Fibonacci sequence is a good example

Ex. , where

The first six terms of this sequence are

given

given
use recursive formula
use recursice formula
use recursive formula

Factorial: If n is a positive integer, n factorial is defined by

****As a special case, zero factorial is defined as ****

Evaluating Factorial Expressions

Ex.

Summation Notation: The sum of the first n terms of a sequence is represented by

where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.

Ex.

Properties of Sums

1. , c is any constant

2.

3.

Arithmetic Sequence: A sequence is arithmetic if the differences between consecutive terms are the same. So, the sequence

is arithmetic if there is a number d such that

The number d is the common difference of the arithmetic sequence.

*The nth term of an arithmetic sequence has the form

where d is the common difference between consecutive terms of the sequence and

The Sum of a Finite Arithmetic Sequence

The sum of a finite arithmetic sequence with n terms is

Finding a Partial Sum of an Arithmetic Sequence

Find the 150th partial sum of the sequence

5, 16, 27, 38, 49....

Solution

For this arithmetic sequence, you have and So,

and the nth term is

.

Therefore, and the sum of the first 150 terms is

Geometric Sequence: A sequence is geometric if the ratios of consecutive terms are the same.

The number r is the common ratio of the sequence

*The nth term of a geometric sequence has the form

where r is the common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in the following form

*The sum of the geometric sequence

with common ratio is

The Sum of an Infinite Geometric Series

If then the infinite geometric series

has the sum

Chapter 12 - Limits

12.1 Introduction to limits!

Definition of Limit - If f(x) becomes abritrarily close to a unique number L as x approaches a from either side, the limit of f(x) as x approaches a is L. This is written as :

It does not matter what happens what x=a specifically, it only matter what is happening as x appraches a.

WHEN LIMITS DO NOT EXIST:

The limis of f(x) as x approaches c does not exist id any of the following conditions is true.

1. f(x) approaches a different number from the right side of c than from the left side of c.

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between two fixed values as x approaches c.

Properties of limits:

12.2 techniques for evalutating limits

Limits of polynomial and rational functions

1. If p is a polynomial function and c is a real numner then the limits of p (x) as x approaches c is p(c)

2. If r is a rational function given by r(x)=p(x)/q(x), and c is a real number such that q(c) doesn't = 0, then the limit of r(x) as x approaches c is = r(c)= p(c)/q(c)

12.4 limits at infinity and limits of sequences

1. "The limit of f(x) as x approaches negative infinity is "

2. "the limit of f(x) as x approaches infinity is "

LIMITS AT INFINITY

for the rational function f(x)=N(x)/D(x), where

and