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9.2 day 2. Photo by Vickie Kelly, 2003. Greg Kelly, Hanford High School, Richland, Washington. Maclaurin Series. Liberty Bell, Philadelphia, PA. Maclaurin Series:. (generated by f at ).

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9.2 day 2

Photo by Vickie Kelly, 2003

Greg Kelly, Hanford High School, Richland, Washington

Maclaurin Series

Liberty Bell, Philadelphia, PA


Maclaurin Series:

(generated by f at )

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.


List the function and its


Evaluate column one

for x = 0.

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.


and substitute for , we get:

If we start with this function:

This is a geometric series with a = 1 and r = -x2.

If we integrate both sides:

This looks the same as the series for sin (x), but without the factorials.


An amazing use for infinite series:

Substitute xi for x.

Factor out the i terms.



This is the series for cosine.

This is the series for sine.

This amazing identity contains the five most famous numbers in mathematics, and shows that they are interrelated.