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9.2 day 2 Finding Common Maclaurin Series Liberty Bell, Philadelphia, PA

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There are some Maclaurin series that occur often enough that they should be memorized. They are on your formula sheet, but today we are going to look at where they come from. Maclaurin Series: (generated by f at )

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List the function and its derivatives.

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List the function and its derivatives. Evaluate column one for x = 0. This is a geometric series with a = 1 and r = x.

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We could generate this same series for with polynomial long division:

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This is a geometric series with a = 1 and r = -x.

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We wouldn’t expect to use the previous two series to evaluate the functions, since we can evaluate the functions directly. We will find other uses for these series, as well. They do help to explain where the formula for the sum of an infinite geometric comes from. A more impressive use of Taylor series is to evaluate transcendental functions.

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Both sides are even functions. Cos (0) = 1 for both sides.

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Both sides are odd functions. Sin (0) = 0 for both sides.

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If we start with this function: and substitute for, we get: This is a geometric series with a = 1 and r = -x 2. If we integrate both sides: This looks the same as the series for sin ( x ), but without the factorials.

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We have saved the best for last!

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INFINITE SEQUENCES AND SERIES

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