SERIES: PART 2

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SERIES PART 2

• Power Series
• Taylor Series
• Maclaurin Series

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Power Series

• Note there is an x value. Usually, there isnt.
  It is usually n for terms.

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Power Series Examples
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Term-by-Term Differentiation

• States that each term within a power series can
  be differentiated term by term at each interior
  point of its interval of convergence

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Term-by-Term Differentiation Examples
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Taylor Series
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Taylor Series Examples
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Maclaurin Series

• The power series representation of a function is
  (x - a) is called Taylor series
• Maclaurin series is similar to Taylor, difference
  is a 0

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Maclaurin Series Examples
EXAMPLE
For the function f (x) sin x, find its first
three Maclaurin polynomials.
SOLUTION
To determine these polynomials, we must first
determine the first, second, and third
derivatives of the function and then evaluate the
function and its three derivatives at x 0.
Now we can find the first three Maclaurin
polynomials.
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Maclaurin Series
CONTINUED
First Maclaurin polynomial
Second Maclaurin polynomial
Third Maclaurin polynomial
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Maclaurin Series Examples