April 25, 2006
Lecturer: Dr Martin Kurth
Trinity Term 2006
Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS)
Problem Sheet 3 – Last Problem Sheet!
This problem sheet is optional.
Due: in the Tutorials 05 May / 08 May
I advise my students to listen carefully the moment they decide to take no more
mathematics courses. They might be able to hear the sound of closing doors.
James Caballero
1. Calculate the limits of the following infinite series, provided they exist:
P∞ ¡ 1 ¢n−1
(a)
,
n=2
P∞ ¡ 2 1 ¢n
,
(b)
n=1 − 4
P∞ n ¡ 1 ¢n−1
(c)
,
n=1 2
4
P∞ n n−1 ¡ 1 ¢n+1
(d)
.
(4 points)
n=1 2 3
8
2. Determine whether the following series are convergent:
P∞ cos2 (n)
(a)
n3 ,
Pn=1
∞ 2−sin n
(b)
,
n=1
n
P∞ e−2n
(c)
n=1 4 ,
P∞ sinh(1/n)
(d)
.
n=1
n2
Hint for (d): You may find it helpful to calculate
d
dx
(4 points)
cosh(1/x).
3. Consider the functions f and g defined by
f (x) = x + 1
and
g(x) = x2 + 1.
Calculate the volume of the solid generated by revolving the area bounded
by the graphs of f and g about the x-axis.
(2 points)
P∞
4. Let n=1 an be convergent, but not absolutely convergent. Show that for
any real number L there exists a rearrangement {bn } of the sequence {an }
such that
∞
X
bn = L.
n=1
(*)
Questions 1, 2 and 3 should be answered by all students, you will get points for them. Question 4 is more challenging and meant as an exercise for the more mathematically interested
students.