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

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10 Oct 2018

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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

1−x=

∞

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.

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.

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).