Taylor and Maclaurin Series

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Lecture 7

• Taylor and Maclaurin Series

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Lecture 7 Objectives

• For a given function, use Taylor's formula, as
  well as power series manipulation, to find
• The Maclaurin Series
• The Taylor Series at a certain value of x
• Find the linearization and the quadratic
  approximation of a function. (See also
  Lecture 1 Objectives.)
• Identify Maclaurin series and find their sums

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An Introduction to Taylor Series

• We saw that power series can sometimes have
  compact sums, e.g.

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Question Given a function f(x), can we find a
power series expansion for it?

• Answer Sometimes.
• Note Since a power series can be differentiated
  term-by-term, we require the condition that
  f(x) is differentiable of all orders.
• Note The above condition is necessary but not
  sufficient.

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Question How do we find the power series
expansion for a function f(x)? (assuming it
exists)

• Answer Easy.
• Starting from f(x) ?ncn(x ? a)n,
• Repeatedly differentiate both sides, and
  substitute x a.
• We then get the coefficients cn f (n)(a)/n!

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Thus, we define
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If we stop the Taylor series at the term n, we
get

• Note The Taylor polynomial Pn(x) plays the role
  of the nth partial sum sn.

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Example Find the Maclaurin series generated by
the function f(x) ex.
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Picture
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Example Find the Taylor series generated by f(x)
ex, at x 2.

• Method 1 Directly using the nth derivative at a
  2.
• Method 2 Write ex e2ex?2, and replace x by (x
  ? 2) in the Maclaurin series for ex.

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Example Find the Taylor series generated by f(x)
cos x, at x 0.
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Picture
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Example Find the Maclaurin series generated by
f(x) sin x.

• Method 1 Directly using the nth derivative.
• Method 2 Take the derivative of the Maclaurin
  series for cos x, and multiply by ?1.

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The Binomial Expansion
Example Find the Maclaurin series generated by
the function f(x) (1 x)m.
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Fast Ways for finding Taylor Series

• Sometimes we can find the required Taylor series
  by manipulating already known series, using
• Addition / Subtraction
• Multiplication / (Long) Division
• Substitution
• Differentiation / Integration
• The interval(s) of convergence are then inherited
  from the original series.

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Example Find the Maclaurin series generated by
f(x) e?x2.

• Method 1 (hard!) Directly using the nth
  derivative.
• Method 2 (Easier) Replace x by ?x2 in the
  Maclaurin series for ex.

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Example Find the Maclaurin series generated by
f(x) cos2x

• Method 1 (hard!) Directly using the nth
  derivative.
• Method 2 (Easier) Use the identity cos2x (1
  cos2x)/2.
• Method 2 (Quick) Square the Maclaurin series for
  cos x (nth term is hard to find).

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Example Find the Maclaurin series generated by
f(x) tan x

• Method 1 (hard!) Directly using the nth
  derivative.
• Method 2 (Quick) Perform a long division of the
  Maclaurin series for (sin x) and (cos x).

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A function without a Taylor Series
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Example Identify the following Maclaurin series
and find their sums
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Answers
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Lecture 7 Objectives (revisited)

• For a given function, use Taylor's formula, as
  well as power series manipulation, to find
• The Maclaurin Series
• The Taylor Series at a certain value of x
• Find the linearization and the quadratic
  approximation of a function. (See also
  Lecture 1 Objectives.)
• Identify Maclaurin series and find their sums

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• Thank you for listening.
• Wafik