Student number
Name [SURNAME(S), Givenname(s)]
MATH 101, Section 212 (CSP)
Week 11: Marked Homework Assignment
Due: Thu 2011 Mar 31 14:00
HOMEWORK SUBMITTED LATE WILL NOT BE MARKED
1. Find a power series representation for the function and determine the interval of convergence.
(a) f (x) =
x3
4x2 +3
(b) f (x) =
x+2
2x2 −x−1
(c) f (x) = ln(3 + x)
(d) f (x) = tan−1 (3x)
(e) f (x) =
2x
(1+x2 )2
(f) f (x) = x sin(x/2)
4
(g) f (x) = e−x + cos(x2 )
2. Evaluate the indefinite integral as a power series. Find the radius of convergence.
(a)
(b)
R
R
tan−1 (x3 ) dx
ex −1
x
dx
3. Use series to approximate the definite integral to within the indicated accuracy. You
may use a calculator.
(a)
(b)
R 0.5
0
R 0.5
0
ln(1 + x5 ) dx
2
x2 e−x dx
with |error| < 5 · 10−7
to within 0.001
4. Evaluate the limit, or determine that the limit does not exist.
x2 −2+2 cos x
x4
(sin x)−x
limx→0 x3
(a) limx→0
(b)
(c) limx→0
(sin x)−x
x4
5. Find the first three nonzero terms in the Maclaurin series for f (x) = 2 sin x cos x, in
three ways:
(a) Use multiplication of Maclaurin series 2(sin x)(cos x)
(b) Use differentiation of the series for sin2 x = 12 (1 − cos 2x)
(c) Use the series for sin 2x
What is f (101) (0)?