Masters’s Exam, Analysis, January 2012
1. Discuss the convergence or divergence of each series Panwhere
(a) an=(−1)n
(n+ 1)7/3
(b) an=ns
nt+ 1,0< s < t
(c) an=n!
2n+4
(d) an=ln(ln(n))
nln(n)
2. Let fn:R→Rand gn:R→Rbe defined for n∈N. Assume, for each n, that both fnand gn
are bounded. Assume the sequences (fn) and (gn) converge uniformly to fand g, respectively.
Show that (fngn) converges uniformly to fg.
3. Let f:R→Ris continuous and let (xn) be a sequence in Rwith limn→xn=A. By directly
using the definition of continuity and the definition of the limit of a sequence (and without
using theorems about limits) prove that
lim
n→∞
f(xn) = f(A).
4. Suppose Ais a compact set in Rmand f:A→Rnis continuous. Show that fis bounded.
5. Let
f(x, y) = (xy2
x2+y2if (x, y)6= (0,0),
0 if (x, y) = (0,0).
Show that fx(0,0) exists but fxis not continuous at the origin.
6. Find and classify the critical points of f(x, y) = xy2(4 −x−y). Then find the maximum value
of fon the closed triangular region in the xy-plane with vertices (0,0),(0,6) and (6,0).
7. Compute the integral ZZR
(x2+y2)3dx dy
where Ris the region in the first quadrant that is bounded by the hyperbolas xy = 1, xy = 3,
x2−y2= 1, x2−y2= 4.
8. Suppose fis integrable over [a, b]. Let c∈(a, b). By using the definition of Riemann-Darboux
integrability rather than by quoting theorems about integrals, show that fis integrable over
[a, c] and
Zb
a
f(x)dx =Zc
a
f(x)dx +Zb
c
f(x)dx
9. Find the Laurent series for the function f(z) = 1
z(2 −z)
(a) in the region |z|>2
(b) in the region |z−1|<1.
10. Compute the integral Z∞
0
cos(x)
(x2+ 1)3dx.