MATH 152 Sample Exam III
SPRING 2016
To be worked in class Tues Apr 26
Part I - Multiple Choice
1. Find the coefficient of (x − 4)3 in the Taylor Series expansion for the function f (x) =
a = 4.
a)
b)
c)
d)
e)
3
256
3
128
1
256
5
264
1
512
2. Descibe the following equation in three dimensional space: x2 + y 2 + z 2 < 6x − 3y + 4
a) A cyliner
b) A circle
3
c) The exterier of the sphere with center 3, − , 0 and radius
2
s
61
4
3
d) The interior of the sphere with center 3, − , 0 and radius
2
s
61
4
3
e) The surface of the sphere with center 3, − , 0 and radius
2
s
61
4
√
x at
(−1)n
.
n4
n=1
Which of the following is true from the Alternating Series Estimation Theorem regarding an
upper bound of the absolute value of the remainder?
1
a) |R4 | <
625
1
b) |R4 | <
256
1
c) |R4 | <
192
1
d) |R4 | <
375
1
e) |R4 | <
132
3. Suppose we use s4 , the fourth partial sum, to approximate the sum of the series S =
4. If
∞
X
cn xn converges at x = 2 and diverges at x = 8, then which of the following is true? Circle
n=1
all true answers.
a)
b)
c)
d)
e)
∞
P
∞
X
cn xn will converge at x = 1
n=1
∞
X
cn xn will converge at x = 3
n=1
∞
X
cn xn will diverge at x = 4
n=1
∞
X
cn xn will diverge at x = 9
n=1
∞
X
cn xn may converge at x = 6.
n=1
5. The series
1
√ is
n
n=2 n −
∞
P
a) divergent because
∞ 1
1
1
P
√ > and
is a divergent p-series.
n− n
n
n=2 n
b) convergent because
c) convergent because
∞ 1
1
1
P
√ < and
is a convergent p-series.
n− n
n
n=2 n
∞
1
1
1
P
√ is a convergent p series.
√ > and
n− n
n
n
n=2
d) divergent by the Ratio Test.
e) divergent by the Test for Divergence.
6. Which of the following series converges?
(I)
∞
X
(−1)n
n=1
n
n+1
a) I only
b) II only
c) III only
d) I and III
e) all 3 series converge
(II)
∞
X
n=1
n!
n1000
(III)
∞
X
(−1)n
2
n=2 n(ln n)
7. Which of the following series is absolutely convergent?
a)
(−1)n
n
n=1
b)
(−1)n
n2
n=1
c)
∞
X
∞
X
∞
X
(−1)n
n=2
d)
e)
1
n ln n
∞
X
1
(−1)n √
n
n=1
∞
X
(−2)n
10
n=1 n + n
8. Find the radius of convergence of the power series
∞
X
(2n + 1)!(2x − 1)n
n=0
a) 0
b) ∞
1
c)
2
1
d) −
2
e) 1
9. Find the interval of convergence of the power series
(2n)!(2x − 1)n
n!
n=0
∞
X
1
2
b) (−∞, ∞)
a)
c) {0}
d) (−1, 1)
e) Diverges everywhere
10. Which of the following is a Maclaurin Series for
a)
(−1)n x4n
(2n)!
n=0
b)
(−1)n x4n+1
n=0 (4n + 1)(2n)!
c)
(−1)n x2n+4
n=0 (2n + 4)(2n + 1)!
d)
(−1)n x4n+3
n=0 (4n + 3)(2n + 1)!
e)
(−1)n x2n+3
n=0 (2n + 3)(2n)!
∞
X
∞
X
∞
X
∞
X
∞
X
Z
x
0
sin(t2 ) dt?
11. Find the second-degree Taylor polynomial of f (x) = e−2x centered at x = −1.
a) T2 (x) = e2 − 2e2 (x + 1) + 2e2 (x + 1)2
b) T2 (x) = e2 − 2e2 (x − 1) + 2e2 (x − 1)2
c) T2 (x) = e2 − 2e2 (x + 1) + 4e2 (x + 1)2
d) T2 (x) = e2 − e2 (x + 1) + e2 (x + 1)2
e) T2 (x) = e2 − 2e2 (x − 1) + 4e2 (x − 1)2
12. Which of the following statements is always true? Circle all that apply.
a) The series
∞
X
(−1)n
n=1
b) The series
converges for all p > 0.
np
∞
X
(−1)n
n=1
np
converges absolutely only if p ≥ 2.
(−1)n
c) The series
converges absolutely by the ratio test
4
n=1 n + n + 1
∞
X
d) If lim an = 0, then
n→∞
∞
X
an converges.
n=1
e) If lim an 6= 0, then
n→∞
∞
X
an diverges.
n=1
13. Find the center and radius of the sphere x2 + y 2 + z 2 + x + 2y − 6z = 0.
√
1
41
a) C( , 1, −3), r =
2
2
√
41
1
b) C(− , −1, 3), r =
2
2
41
1
c) C(− , −1, 3), r =
2
2
1
41
d) C(− , 1, −3), r =
2
2
√
41
1
e) C(− , 1, −3), r =
2
2
Part II - Work Out Problems
14. Find the radius and interval of convergence of the power series
∞
X
(−1)n (x − 3)n
n2n
n=1
15. Find a power series for g(x) =
Z
x3 ln(1 + x2 ) dx.
16. Find the power series about 0 for f (x) =
2x
(1 − x2 )2
17. Find the Taylor Series for f (x) =
1
centered at a = 1.
x+1
18. Prove the series
∞
X
3
n2 e−n converges. Use the sum of the first 4 terms to approximate the sum
n=1
of the series and estimate the error.
19. Approximate the series
∞
X
1
1
with error less than
.
3
30
n=1 n