APPM 1360 Midterm: 1360exam3_spring2014
APPM 1360 Exam 3 Spring 2014
INSTRUCTIONS: Books, notes, and electronic devices are not permitted. Write (1) your name, (2) 1360/EXAM 3,
(3) lecture number/instructor name and (4) SPRING 2014 on the front of your bluebook. Also make a grading table
with room for 4 problems and a total score. Start each problem on a new page. Box your answers. A correct answer
with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial
credit. SHOW ALL WORK! JUSTIFY ALL YOUR ANSWERS!
1. The following parts are not related, justify your answers:
(a) (8 pts) For what values of x(if any) is the power series
∞
X
n=0
(−1)nx2n+1
(n+ 1)4nabsolutely convergent? conditionally
convergent?
(b) (8 pts) Suppose
∞
X
n=0
cnxnconverges when x=−4 and diverges when x= 6, what can be said about the absolute
convergence, conditional convergence or divergence of
∞
X
n=106
cn3n
2?
(c) (8 pts) Find the sum of the series
∞
X
n=1
2n1
2n
. (Hint: Differentiate a geometric series.)
2. (a) (7 pts) Find a Maclaurin series of f(x) =
sin(3x)
3x, x 6= 0
1, x = 0
. Express your answer in Σ-notation. (Hint: Refer to
the formula sheet.)
(b) (7 pts) Use your answer for part (a) to find a Maclaurin series for f′(x). Express your answer in Σ-notation.
(c) (7 pts) Estimate the value of f′(1/3) using T4(x), the 4th order Taylor polynomial of f′(x).
(d) (7 pts) Use the Alternating Series Estimation Theorem to determine the fewest number of nonzero terms needed to
ensure that your approximation in part (c) is accurate to within 0.01.
3. The following parts are not related, justify your answers:
(a) (8 pts) Suppose f(x) can be represented as a Maclaurin Series and suppose if −1≤x≤1 then |f(n)(x)|<|cos(nx)|
1 + x2,
for n= 0,1,2, . . . . Find the best possible error bound if we approximate f(0.5) using T3(x), the 3rd-order Taylor
Polynomial (assume that T3(x) is centered at a= 0).
(b) (8 pts) Evaluate the limit using Taylor Series: lim
t→0
3 tan−1(t)−3t+t3
t5
(c) (8 pts) Suppose f(x) = 5
√x+ 2, find f(21)(0). Show all work, you don’t need to simplify any binomial coefficients.
4. Determine if the following are absolutely convergent, conditionally convergent or divergent, justify your answers:
(a)(8 pts)
∞
X
n=0
(−1)n
4
√n2+n+ 4 (b)(8 pts)
∞
X
k=8
1
ln(kk)(c)(8 pts)
∞
X
n=1
nn
4n(n+ 1)!
— FORMULA SHEET ON THE OTHER SIDE —
Document Summary
Instructions: books, notes, and electronic devices are not permitted. Write (1) your name, (2) 1360/exam 3, (3) lecture number/instructor name and (4) spring 2014 on the front of your bluebook. Also make a grading table with room for 4 problems and a total score. A correct answer with incorrect or no supporting work may receive no credit, while an incorrect answer with relevant work may receive partial credit. Justify all your answers: the following parts are not related, justify your answers: (a) (8 pts) for what values of x (if any) is the power series. Xn=0 ( 1)nx2n+1 (n + 1)4n absolutely convergent? conditionally convergent? (b) (8 pts) suppose. Xn=0 cnxn converges when x = 4 and diverges when x = 6, what can be said about the absolute convergence, conditional convergence or divergence of. 2 (c) (8 pts) find the sum of the series.
