9 2 day 2 Finding Common Maclaurin Series

9. 2 day 2 Finding Common Maclaurin Series Liberty Bell, Philadelphia, PA Photo by Vickie Kelly, 2003 Greg Kelly, Hanford High School, Richland, Washington

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 )

List the function and its derivatives.

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

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

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

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

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

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

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.

We have saved the best for last!

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