I had an email last week from a teacher asking, how come I can use a substitution to find a power series for , and for , but not for ?

The answer is that you can. Substituting (2x) into the cosine’s series give you a Taylor series centered at *x* = 0, a Maclaurin Series. Substituting (*x* – 1) into the series for *e*^{x} gives you a Taylor series centered at *x* = 1. And substituting into the cosine series gives you a Taylor series centered at . I suspect that she was hoping for or was asked to find a Maclaurin series, not one with such a strange center.

The center of a Taylor series is the value of *x* that makes its argument zero.

**AP Exam Question 2004 BC 6(a)**

This brought to mind the AP Exam question 2004 BC 6(a) where students were asked to write the third-degree Taylor polynomial about *x* = 0 for the function . The intended method was for students to find the first three derivative and substitute them into the general form for a Taylor series. That’s what students who got this correct did. This is the only time I can remember when students were expected to do that; usually they manipulate a given series or substitute into a known series.

A number of students tried to substitute into the series for the sine. This gets a very nice Taylor series centered at . This earned no credit since a Maclaurin series was required.

But there is another way! (I originally wrote, “But there is an easier way!” but it’s only easier if you see how to do it.)

**Trigonometry to the Rescue!**

Then using the first two terms each from the series for sine and cosine you get the correct answer:

This brings us to , which can be approached the same way. Here is the entire Maclaurin series.

**Moral:** Trig can be very useful.

Here is a previous post, Geometric Series – Far Out, that shows a “mistake” you may find interesting.

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