Taylor_and_Maclaurin_Series.ppt

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TAYLOR AND MACLAURIN

• how to represent certain types of functions as
  sums of power series

• You might wonder why we would ever want to
  express a known function as a sum of infinitely
  many terms.

• Integration. (Easy to integrate polynomials)
• Finding limit
• Finding a sum of a series (not only geometric,
  telescoping)

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TAYLOR AND MACLAURIN
Example
Maclaurin series ( center is 0 )
Example
Find Maclaurin series
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TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of
Convergence
MEMORIZE these Maclaurin Series
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TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example
Find Maclaurin series
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TAYLOR AND MACLAURIN
TERM-081
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TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
TERM-101
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TAYLOR AND MACLAURIN
TERM-082
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Sec 11.9 11.10 TAYLOR AND MACLAURIN
TERM-102
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Sec 11.9 11.10 TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Example
Find the sum of the series
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TAYLOR AND MACLAURIN
TERM-102
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TAYLOR AND MACLAURIN
TERM-082
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TAYLOR AND MACLAURIN
Example
Find the sum
Leibnizs formula
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TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of
Convergence
MEMORIZE these Maclaurin Series
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The Binomial Series
DEF
Example
Example
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The Binomial Series
binomial series.
NOTE
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The Binomial Series
TERM-101
binomial series.
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The Binomial Series
TERM-092
binomial series.
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TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of
Convergence
Example
Find Maclaurin series
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TAYLOR AND MACLAURIN
TERM-102
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TAYLOR AND MACLAURIN
TERM-111
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TAYLOR AND MACLAURIN
TERM-101
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TAYLOR AND MACLAURIN
TERM-082
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TAYLOR AND MACLAURIN
Important Maclaurin Series and Their Radii of
Convergence
MEMORIZE these Maclaurin Series
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TAYLOR AND MACLAURIN
Maclaurin series ( center is 0 )
Taylor series ( center is a )
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TAYLOR AND MACLAURIN
TERM-091
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TAYLOR AND MACLAURIN
TERM-092
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TAYLOR AND MACLAURIN
TERM-082
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TAYLOR AND MACLAURIN
Taylor series ( center is a )
DEF
Taylor polynomial of order n
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TAYLOR AND MACLAURIN
The Taylor polynomial of order 3 generated by the
function f(x)ln(3x) at a1 is
TERM-102
DEF
Taylor polynomial of order n
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TAYLOR AND MACLAURIN
TERM-101
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TAYLOR AND MACLAURIN
TERM-081
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TAYLOR AND MACLAURIN
Taylor series ( center is a )
Taylor polynomial of order n
Remainder
Taylor Series
Taylors Formula
Remainder consist of infinite terms
for some c between a and x.
REMARK
Observe that
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TAYLOR AND MACLAURIN
Taylors Formula
for some c between a and x.
Taylors Formula
for some c between 0 and x.
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TAYLOR AND MACLAURIN
Taylor series ( center is a )
DEF
nth-degree Taylor polynomial of f at a.
DEF
Remainder
Example
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TAYLOR AND MACLAURIN
TERM-092
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TAYLOR AND MACLAURIN
TERM-081