Engineering Math II – Spring 2015
Name:
Section: 561 / 562 / 563
Directions. Read each problem carefully and work it out in the space provided. Put a
box around your answer to mark it clearly. Turn in your work with this sheet of paper
stapled on top.
P∞
n
Problem 1. If
n=0 cn 4 is convergent, does it follow that the following series are
convergent? Explain your answer.
∞
X
(i)
cn (−2)n
n=0
(ii)
∞
X
cn (−4)n
n=10
Problem 2. Find the radius of convergence and interval of convergence of the series.
(a)
∞
X
(−1)n xn
n=1
(b)
x1/3
∞
X
n2 xn
n=0
10n
MATH 152:561-563 – Spring 2015
(c)
Quiz # 8
2
∞
X
(−1)n x2n−1
n=1
(2n − 1)!
Problem 3. Find a power series representation for the function and determine the radius
of convergence.
(a)
(b)
x4
1
+ 16
1 + x2
1 − x2
(c) x ln(1 + x)
MATH 152:561-563 – Spring 2015
Quiz # 8
3
Problem 4. Find the Taylor series for f (x) at the given value of a. Assume that f has
a power series expansion. Do not show that Rn (x) → 0.
(a) f (x) = ln(x);
(b) f (x) =
√
x;
(c) f (x) = cos(x);
a=2
a=4
a=0
MATH 152:561-563 – Spring 2015
Quiz # 8
4
Problem 5. Use the McLaurin series for cos(x) obtained in part (c) of Problem 4 to find
the McLaurin series for x2 cos(3x).
Z
Problem 6. Evaluate the indefinite integral
x
dx as a power series.
1 + x5
MATH 152:561-563 – Spring 2015
Quiz # 8
5
Problem 7. Find the McLaurin series of f (by any method) and its radius of convergence.
(a) f (x) = √
(b) f (x) = 2x
1
1 + 2x