UNIVERSITY OF DUBLIN
XMA1213
TRINITY COLLEGE
Faculty of Science
school of mathematics
Hilary Term 2008
JF Maths, JF TP
JF TSM, SF TSM
Course 121
Monday, March 10
Regent House
14:00 – 17:00
Dr. P. Karageorgis
Attempt all questions. All questions are weighted equally.
You may use non-programmable calculators, but you may not use log tables.
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 2008
°