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Section 8.6/8.7: Taylor and Maclaurin Series Practice HW from Stewart Textbook (not to hand in) p. 604 # 3-15 odd, 21-27 odd p. 615 # 5-25 odd, 31-37 odd

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Taylor Series In this section, we discuss how to use a power series to represent a function. Definition: If has a power series representation, then and which is called a Taylor series at x = a.

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If a = 0, then is the Maclaurin series of f centered at x = 0.

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Example 1: Find the Maclaurin series of the function. Find the radius of convergence of this series. Solution:

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Example 2: Find the Maclaurin series of the function. Find the radius of convergence of this series. Solution: (In typewritten notes)

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Note Once we know the Maclaurin (Power) series representations centered at x = 0 for a given function, we can find the Maclaurin (Power) series of other functions by substitution, differentiation, or integration.

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Some Common Maclaurin Series Series Interval of Convergence

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Example 3: Use a known Maclaurin series to find the Maclaurin series of the given function. Solution:

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Example 4: Use a known Maclaurin series to find the Maclaurin series of the given function. Solution:

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Note: To differentiate or integrate a Maclaurin or Taylor series, we differentiate or integrate term by term.

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Example 5: Find the Maclaurin series of Solution:

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Example 6: Use a series to estimate to 3 decimal places. Solution: (In typewritten notes)

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Example 7: Find the Taylor series of at. Solution:

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Example 8: Find the Taylor series of at. Solution:

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