Infinite Series

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Infinite Series

• Dr. Ching I Chen

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9.1 Power Series (1)Geometric Series
Example
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9.1 Power Series (2)Geometric Series

• The Partial sums of the series form a sequence
• ( s1, s2, s3, s4, )

of real numbers, each defined as a finite sum.
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9.1 Power Series (3)Geometric Series
Example
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9.1 Power Series (4)Geometric Series
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9.1 Power Series (5, Example 1)Geometric Series
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9.1 Power Series (6, Example 2)Geometric Series
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9.1 Power Series (7)Geometric Series

• The series is a geometric series if each term is
  obtained from its preceding term by multiplying a
  same number r.

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9.1 Power Series (8, Example 3)Geometric Series
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9.1 Power Series (9)Representing Function by
series

• The series is not only a number series but also
  is a form of function x series.

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9.1 Power Series (10)Representing Function by
series
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9.1 Power Series (11)Representing Function by
series
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9.1 Power Series (12, Exploration
1-1)Representing Function by series
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9.1 Power Series (13, Exploration
1-2)Representing Function by series
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9.1 Power Series (14, Exploration
1-3)Representing Function by series
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9.1 Power Series (15, Exploration
1-4)Representing Function by series
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9.1 Power Series (16, Exploration
1-5)Representing Function by series
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9.1 Power Series (17)Differentiation and
Integration

• The geometric function series is the form of
  polynomials. Therefore, the application of
  differentiation can be used to find a function
  series.

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9.1 Power Series (18, Example 4)Differentiation
and Integration
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9.1 Power Series (19, Example 5)Differentiation
and Integration
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9.1 Power Series (20, Theorem 1)Differentiation
and Integration
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9.1 Power Series (21)Differentiation and
Integration
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9.1 Power Series (22, Theorem 2)Differentiation
and Integration
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9.1 Power Series (23)Differentiation and
Integration

• Finding a power series by integration

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9.1 Power Series (24, Exploration
2-1)Differentiation and Integration
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9.1 Power Series (25, Exploration
2-2)Differentiation and Integration
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9.1 Power Series (26, Exploration
2-3)Differentiation and Integration
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9.1 Power Series (27, Exploration
2-4)Differentiation and Integration
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9.1 Power Series (28, Exploration 3)Identifying
a Series
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9.2 Taylor series (1)Constructing a Series
If the specified function is given, Then, is it
possible to find the polynominal based on x 0 ?
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9.2 Taylor series (2, Example 1)Constructing a
Series
What is its polynomial ?
This is called the fourth order Taylor polynomial
for the function ln(1x) at x 0.
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9.2 Taylor series (3, Example 2-1)Series for
sin x and cos x
This is called the Taylor series generated by the
function sin(x) at x 0.
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9.2 Taylor series (4, Example 2-2)Series for
sin x and cos x

• Constructing a power series for sin x at x 0

The higher order of polynomial taken, the more
accurate of the function reached.
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9.2 Taylor series (5, Exploration 2-1)Series
for sin x and cos x
This is called the Taylor series generated by the
function cos(x) at x 0.
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9.2 Taylor series (6, Exploration 2-2)Series
for sin x and cos x

• Constructing a power series for cos x at x 0

The higher order of polynomial taken, the more
accurate of the function reached.
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9.2 Taylor series (7)Beauty Bare
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9.2 Taylor series (8)Maclaurin and Taylor Series
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9.2 Taylor series (9, Example 3)Maclaurin and
Taylor Series
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9.2 Taylor series (10, Exploration 3)Maclaurin
and Taylor series
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9.2 Taylor series (11)Maclaurin and Taylor
Series
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9.2 Taylor series (12, Example 4)Maclaurin and
Taylor Series
This is called the Taylor series generated by the
function ex at x 2.
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9.2 Taylor series (13, Example 5)Maclaurin and
Taylor Series
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9.2 Taylor series (14)Table of Maclaurin Series

• Taylor series (Maclaurin series) for several
  functions

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9.2 Taylor series (15)Table of Maclaurin Series

• Taylor series can be added, subtracted, and
  multiplied, and multiplied by constants and
  powers of x, and the results are once again
  Taylor series.

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9.3 Taylor Theorem (1)About Taylor Polynomials

• A function can be expressed as Taylor series
  (infinite terms) or Taylor Polynomial (finite
  terms). However, the more terms of the
  Polynomial, the more accuracy of the function can
  be reached.

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9.3 Taylor Theorem (2, Example 1)About Taylor
Polynomials
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9.3 Taylor Theorem (3, Example 2)About Taylor
Polyniminals
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9.3 Taylor Theorem (4, Theorem 3)The Remainder
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9.3 Taylor Theorem (5, Example 3)The Remainder
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9.3 Taylor Theorem (6, Exploration 1)The
Remainder
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9.3 Taylor Theorem (7, Theorem 4)Remainder
Estimation Theorem
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9.3 Taylor Theorem (8, Example 4)Remainder
Estimation Theorem
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9.3 Taylor Theorem (9, Example 5)Remainder
Estimation Theorem
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9.4 Radius of Convergence (1) Convergence
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9.4 Radius of Convergence (2) Convergence
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9.4 Radius of Convergence (3, Example 1)
Convergence
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9.4 Taylor Theorem (4, Theorem 5) Radius of
converge
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9.4 Taylor Theorem (5, Theorem 6) N-term Test
for Converge
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9.4 Taylor Theorem (6, Example 2) N-term Test
for Converge
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9.4 Taylor Theorem (7, Theorem 7) Comparing
Nonnegtive Series
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9.4 Taylor Theorem (8, Example 3) Comparing
Nonnegtive Series
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9.4 Taylor Theorem (9, Theorem 8) Comparing
Nonnegtive Series
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9.4 Taylor Theorem (10, Example 4) Comparing
Nonnegtive Series
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9.4 Taylor Theorem (11) Ratio Test
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9.4 Taylor Theorem (12, Exploration 1) Ratio
Test
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9.4 Taylor Theorem (13, Example 5) Ratio Test
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9.4 Taylor Theorem (14, Example 6) Ratio Test
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9.5 Testing Convergence at Endpoints (1)
Integral Test (Theorem 10)
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9.5 Testing Convergence at Endpoints (2)
Integral Test (Example 1)
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9.5 Testing Convergence at Endpoints (3)
Harmonic Series and p-series

• When p 1 is called the harmonic series. It is
  probably the most famous divergent series in
  mathematics.

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9.5 Testing Convergence at Endpoints (4)
Integral Test (Example 2)
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9.5 Testing Convergence at Endpoints (5)
Comparison Tests (Theorem 11)
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9.5 Testing Convergence at Endpoints (6)
Comparison Tests (Example 3)
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9.5 Testing Convergence at Endpoints (7)
Alternating Series

• A series in which the terms are alternately
  positive and negative is an alternating series

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9.5 Testing Convergence at Endpoints (8)
Alternating Series (Theorem 12)
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9.5 Testing Convergence at Endpoints (9)
Alternating Series

• The partial sums keep overshooting the limit as
  they go back and forth on the number line,
  gradually closing in as the terms tend to zero.
  If we stop at the nth partial sum, we know that
  the next term (un1) will be again cause us to
  overshooting the limit in the positive or
  negative direction, depending on the sign carried
  by un1.

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9.5 Testing Convergence at Endpoints (10)
Alternating Series (Theorem 13)
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9.5 Testing Convergence at Endpoints (11)
Alternating Series (Example 4)
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9.5 Testing Convergence at Endpoints (12)
Absolute and Conditional Convergence
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9.5 Testing Convergence at Endpoints (13)
Absolute and Conditional Convergence (Example 5)
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9.5 Testing Convergence at Endpoints (14)
Intervals of Convergence
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9.5 Testing Convergence at Endpoints (15)
Intervals of Convergence
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9.5 Testing Convergence at Endpoints (16)
Intervals of Convergence (Example 6)
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9.5 Testing Convergence at Endpoints (17)
Procedure for Convergence
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