UNIVERSITY OF DUBLIN
XMA1213
TRINITY COLLEGE
Faculty of Engineering, Mathematics
and Science
school of mathematics
JF Maths, JF TP
JF TSM
Hilary Term 2009
Course 121
Monday, March 9
Regent House
14:00 – 16:00
Dr. P. Karageorgis
Attempt all questions. All questions are weighted equally.
Log tables are available from the invigilators, if required.
Page 2 of 2
1. Compute each of the following integrals:
Z
3x + 5
dx,
x3 − x
XMA1213
Z
x cos x dx.


 1 if x 6= 0 
2. Let f be defined by f (x) =
. Show that f is integrable on [0, 1].
 0 if x = 0 
3. Define a sequence {an } by setting a1 = 4 and
an+1 =
1
5 − an
for each n ≥ 1.
Show that 0 ≤ an+1 ≤ an ≤ 4 for each n ≥ 1, use this fact to conclude that the sequence
converges and then find its limit.
4. Compute each of the following limits:
lim
x→0
ex − x − 1
,
x2
lim x sin(1/x).
x→∞
5. Test each of the following series for convergence:
∞
X
(−1)n−1 n2
n=1
1 + n4
,
¶n
∞ µ
X
2n
.
1
+
3n
n=1
6. Find the radius of convergence of the power series
∞
X
(n!)2 n
f (x) =
·x .
(2n)!
n=0
7. Suppose f is a differentiable function such that f 0 (x) = 2x · f (x) for all x ∈ R. Show that
2
there exists some constant C such that f (x) = Cex for all x ∈ R.
8. Suppose that f is a function with
|f (x) − f (y)| ≤ |x − y|2
for all x, y ∈ R.
Using the limit definition of the derivative, show that f is actually constant.
c UNIVERSITY OF DUBLIN 2009
°