Maclaurin and Taylor Series

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Maclaurin and Taylor Series

• Review
• We defined
• The nth Maclaurin polynomial for a function f as
• The nth Taylor polynomial for f about x x0 as

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Maclaurin and Taylor Series
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Example

• Find the Maclaurin series for
• (a) (b)
  (c) (d)
• Solution (a) We take the Maclaurin polynomial
  and extend it

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• (b) We take the Maclaurin polynomial and
  extend it.
• (c)

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Example

• Find the Taylor series for 1/x about x 1.
• In the previous lecture we have found the nth
  Taylor polynomial for 1/x about x 1 is
• Thus, the Taylor series for 1/x about x 1 is

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Power Series in x

• If are constants and x is a
  variable, then a series of the form
• is called a power series in x. Some examples are

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Radius and Interval of Convergence

• If a numerical value is substituted for x in a
  power series then the
  resulting series of numbers may either converge
  or diverge. This leads to the problem of
  determining the set of x-values for which a given
  power series converges this is called its
  convergence set.
• Observe that every power series in x converges at
  x 0. In some cases, this may be the
  only number in the convergence set. In other
  cases the convergence set is some finite or
  infinite interval containing x 0.

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Radius and Interval of Convergence
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Radius and Interval of Convergence
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Example

• Find the interval of convergence and radius of
  convergence of the following power series.
• (a) (b)
  

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Power Series in x x0

• If xo is a constant and if x is replaced by x
  xo in the power series expansion, then the
  resulting series has the form
• This is called a power series in x xo. Some
  examples are
• More generally, the Taylor series
• is a power series in x xo.

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Radius and interval of convergence of a Power
Series in x x0
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Radius and interval of convergence of a Power
Series in x x0
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Example

• Find the interval of convergence and radius of
  convergence of the series

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