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AP Calculus Question of the Month: June
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by Lin McMullin
Educational Consultant and Writer
Niantic, Connecticut
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Neighborhoods of Infinity
Let's begin with a warm-up problem before we get to the Question of the Month:
Write the first four terms and the general term of the power series centered at
(the Maclaurin Series) for the function 
. What is the interval of convergence? Graph the function and its Taylor Polynomial of
degree 5.
Stop here and work the problem.
There are a variety of ways of doing this problem. One of the easiest is to consider the formula for the sum of a geometric series with a first term of a and a common ratio of r, namely 
. Our function looks like this one with
and
. So,
Geometric series converge when
or
. So the interval of convergence is
.
(in black) and the fifth degree Taylor Polynomial
(in blue). As you can see, in the interval of convergence, the function and its Taylor Polynomial are very close to the same. -->
So far, so good (and soooo ordinary). However, one teacher was lucky enough to have a student who changed the expression by dividing everything by 2x and came up with this:
Well, this also looks like a geometric series, this time with
. So here is the Question of the Month for you:
(A) Using
, write the first five terms and the general term of the geometric series for
.
(B) What is the interval of convergence?
(C) Graph the original function and the first five terms of this series on the same axes and compare the graphs. Discuss what you see.
(D) Is the series a power series? Explain. |
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Lin McMullin, an educational consultant and writer with extensive experience teaching AP Calculus, lives in Niantic, Connecticut. As a College Board consultant he has presented AP Calculus institutes and workshops in the United States and Europe, and is a Table Leader at the Reading. His work as a writer includes co-authorship of the popular D&S Marketing Systems review books for the AP Calculus Exam and Teaching AP®
Calculus, a book especially for AP Calculus teachers.
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