CAS MA 124 Study Guide - Midterm Guide: Ion, Binomial Series

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10 Oct 2018
Professor
MA 124 - Fall ’12 - Exam 3A
GENERAL INSTRUCTIONS: Show all your work. A correct answer without justifi-
cation receives no credit. Check your algebra and arithmetic carefully.
1) (16 points)
a) (10 points) Use the fact that
1
1x=
X
k=0
xkfor |x|<1
to find a power series for
f(x) = 1
(1 x)2
b) (6 points) What is the interval of convergence of the series you obtained in a) ? Justify
your answer.
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2) (16 points) Use the binomial series
(1 + x)p=
X
k=0 p
kxkfor |x|<1
to find an infinite series that gives the exact value of
Z0.5
0
1
1 + x6dx .
Write out the first three non-zero terms of this series.
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3) (16 points)
a) (4 points) Find the Taylor polynomial p2(x) of degree 2 and center 0 for the function
f(x) = ln(1 + x)
b) (4 points) Use p2(x) to write an approximate value for
ln 1.3
c) (4 points) Write the remainder term R2(x) for f(x) = ln(1 + x) and a= 0.
d) (4 points) Use your expression in c) to estimate the maximum absolute error of your
approximation in b).
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