Sequences and Series

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Chapter 10

• Sequences and Series

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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.

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Example
nth term formula
Sequence
First Term
Second Term
Fourth Term
nth Term
Third Term
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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
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Partial Sums of a Sequence

• For the sequence
• the partial sums are

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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.

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Example
End with 5
Start with 1
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Properties of Sums
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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

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Partial Sums of Arithmetic Sequences

• For the arithmetic sequence,
• The nth partial sum
• Is given by either of the following formulas.
• 1.
• 2.

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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

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Partial Sums of Geometric Sequences

• For the geometric sequence
• the nth partial sum
• is given by

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Sum of an Infinite Geometric Series

• A sum of the form
• is called an infinite series.
• If then the infinite series
• has the sum

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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
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• 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
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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.
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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
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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.
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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(k1) is true.
• Then P(n) is true for all natural number n.

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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(k1) is
  true.

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• Prove For all natural numbers n,

Step 1 Show P(1) true.
Step 2 Assume P(k) is true.
Step 3 Show P(k1) is true.
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Manipulate the left hand side so it looks like
the right hand side.
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We have now shown that if P(k) is true, P(k1) is
also true. The induction step is completed.
Hence, by the Principle of Mathematical
Induction, for all natural numbers n,