Engineering Math II – Spring 2015
Quiz #6
Name:
Section: 561 / 562 / 563
Part I: Multiple Choice. Read each problem carefully and work it out on a separate
sheet of paper. Put a box around the choice you believe best answers the question. Turn
in your work with this sheet of paper stapled on top.
Problem 1 (2 pts). If the integral test is used to determine whether the series
converges, we find that
Z ∞
P
2
2
−n2
xe−x dx = , hence ∞
converges.
(a)
n=1 ne
e
1
Z ∞
P
e
2
−n2
(b)
xe−x dx = , hence ∞
converges.
n=1 ne
2
1
Z ∞
P
2
−n2
xe−x dx = 0, hence ∞
converges.
(c)
n=1 ne
P∞
1
Z
∞
2
xe−x dx =
(d)
1
Z
(e)
∞
2
P
1
−n2
converges.
, hence ∞
n=1 ne
2e
xe−x dx = ∞, hence
P∞
n=1
2
ne−n diverges.
1
√
∞
X
n
Problem 2 (2 pts). The series
2
n +1
n=1
√
∞
X
n
1
1
(a) diverges because 0 ≤ 2
≤ 3 and
3 diverges by p-series.
n +1
2
n2
n
n=1
√
∞
X
n
1
1
(b) converges because 0 ≤ 2
≤ 3 and
3 converges by p-series.
n +1
2
n2
n
n=1
√
∞
X
n
1
1
(c) converges because 0 ≤ 2
≤ 2 and
converges by p-series.
2
n +1
n
n
n=1
√
∞
X
1
n
1
and
diverges by p-series.
(d) diverges because 0 ≤ ≤ 2
n
n +1
n
n=1
√
n
(e) converges because lim 2
= 0.
n→∞ n + 1
n=1
2
ne−n
MATH 152:561-563 – Spring 2015
Quiz #6
Problem 3 (2 pts). What does it mean to say
2
∞
X
an exists?
n=1
(a) lim an = 0.
n→∞
(b) lim an exists.
n→∞
(c) There are numbers A and B such that A ≤ an ≤ B for all n.
(d) There are numbers A and B such that A ≤ a1 + a2 + · · · + an ≤ B for all n.
(e) lim (a1 + a2 + · · · + an ) exists.
n→∞
Problem 4 (2 pts). The sequence defined by a1 = 2 and an+1 = 5 − a4n is increasing and
bounded above. This sequence
(a) diverges.
(b) converges to 1.
(c) converges to 5.
(d) converges to 4.
(e) converges to 3.
MATH 152:561-563 – Spring 2015
Quiz #6
3
Part II: Free response. Read each question carefully. Work out the problem in the
space provided. Mark your answer clearly by enclosing it in a circle or a box. You will
be graded both on the correctness of your answer and the quality of your work.
Problem 5. Consider the series
1
∞
X
e− n
n=1
n2
.
(a) (3 pts) Use the Integral Test to prove convergence.
(b) (3 pts) Estimate the error if we approximate the series by the 9th partial sum.
MATH 152:561-563 – Spring 2015
Problem 6. Consider the series S =
Quiz #6
∞
X
n=2
4
1
.
n(ln(n))4
(a) (3 pts) Does this series converge or diverge? Explain your answer.
(b) (3 pts) Find s6 , that is the sum of the first five terms of the series, to approximate
the sum of the series and find a bound of the remainder.