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Chapter 10. Sequences and Series. 10.1 Sequences and Summation Notation. Definition: A sequence is a function f whose domain is the set of natural numbers. The values are called terms of the sequence. Example:. nth term formula. Sequence. First Term. Second Term. Fourth Term.

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chapter 10

Chapter 10

Sequences and Series

10 1 sequences and summation notation
10.1 Sequences and Summation Notation

Definition:A sequence is a function f whose domain is the set of natural numbers. The values are called terms of the sequence.

slide3

Example:

nth term formula

Sequence

First Term

Second Term

Fourth Term

nth Term

Third Term

recursive sequences
Recursive Sequences

Definition:A sequence that is defined by listing the first term, or the first few terms, and then describes how to determine the remaining terms from the given ones is recursive.

Example:

partial sums of a sequence
Partial Sums of a Sequence

For the sequence

the partial sums are

sigma notation
Sigma Notation

Given

we can write the sum of the first n terms using summation notation, or sigma notation.

k is called the index of summation, or summation variable.

example
Example

End with 5

Start with 1

10 2 arithmetic sequences
10.2 Arithmetic Sequences

Definition:An arithmetic sequence is a sequence of the form

The number a is the first term, and d is the common difference of the sequence. The nth term of an arithmetic sequence is given by

partial sums of arithmetic sequences
Partial Sums of Arithmetic Sequences

For the arithmetic sequence,

The nth partial sum

Is given by either of the following formulas.

1.

2.

10 3 geometric sequences
10.3 Geometric Sequences

Definition: A geometric sequence is a sequence of the form

The number a is the first term, and r is the common ratio of the sequence. The nth term of an arithmetic sequence is given by

partial sums of geometric sequences
Partial Sums of Geometric Sequences

For the geometric sequence

the nth partial sum

is given by

sum of an infinite geometric series
Sum of an Infinite Geometric Series

A sum of the form

is called an infinite series.

If then the infinite series

has the sum

10 5 mathematical induction
10.5 Mathematical Induction

Consider the following sums. Look for a pattern.

The sum of the first 1 odd number is

The sum of the first 2 odd numbers is

The sum of the first 3 odd number is

The sum of the first 4 odd number is

The sum of the first 5 odd number is

slide15
Can we make a conjecture about the sum of the first n odd integers?

The sum of the first n odd integers is

What is the form of any odd integer?

Mathematically, our conjecture now reads …

can we prove it
Can we prove it?

A proof is a clear argument that demonstrates the truth of a statement beyond doubt.

To prove our conjecture, we will use a special kind of proof called mathematical induction.

mathematical induction
Mathematical Induction

To prove something using induction, we need to establish a sequence of mathematical statements. We call these statements P1, P2, P3, etc.

The sum of the first 1 odd number is

The sum of the first 2 odd numbers is

The sum of the first 3 odd number is

the key idea
The Key Idea

Suppose we can prove that whenever one of these statements is true, the statement following it is also true.

For every k, if is true, then is true.

This is called the induction step.

principle of mathematical induction
Principle of Mathematical Induction

For each natural number n, let P(n) be a statement depending on n. Suppose the following two conditions are satisfied.

  • P(1) is true.
  • For every natural number k, if P(k) is true, then P(k+1) is true.

Then P(n) is true for all natural number n.

how it works
How It Works.
  • Show P(1) is true. The induction step then leads through if P(1) is true, then P(2) is true. If P(2) is true, then P(3) is true, etc.
  • Assume P(k) is true.
  • Use P(k) (and some algebra) to show P(k+1) is true.
slide21
Prove: For all natural numbers n,

Step 1: Show P(1) true.

Step 2: Assume P(k) is true.

Step 3: Show P(k+1) is true.

slide23

We have now shown that if P(k) is true, P(k+1) is also true. The induction step is completed.

Hence, by the Principle of Mathematical Induction, for all natural numbers n,