MATH 409–501/503
Fall 2013
Sample problems for the final exam
Any problem may be altered or replaced by a different one!
Problem 1 (20 pts.) Suppose E1 , E2 , E3 , . . . are countable sets. Prove that their union
E1 ∪ E2 ∪ E3 ∪ . . . is also a countable set.
Problem 2 (20 pts.) Find the following limits:
√
x−8
1
c n
√
(i) lim log
,
(ii)
lim
, where c ∈ R.
,
(iii)
lim
1
+
x→0
x→64 3 x − 4
n→∞
1 + cot(x2 )
n
Problem 3 (20 pts.) Prove that the series
∞
X
(−1)n+1
n=1
x3 x5 x7
x2n−1
= x−
+
−
+ ...
(2n − 1)!
3!
5!
7!
converges to sin x for any x ∈ R.
Problem 4 (20 pts.) Find an indefinite integral and evaluate definite integrals:
√
Z p
Z ∞
Z √3 2
x +6
1+ 4x
√
(i)
dx,
(iii)
x2 e−x dx.
dx,
(ii)
2 +9
x
2 x
0
0
Problem 5 (20 pts.)
For each of the following series, determine whether the series
converges and whether it converges absolutely:
√
∞ √
∞ √
∞
X
X
X
n+1− n
n + 2n cos n
(−1)n
√
(ii)
(i)
,
(iii)
.
√ ,
n!
n
log
n
n
+
1
+
n
n=1
n=1
n=2
Bonus Problem 6 (15 pts.) Prove that an infinite product
∞
Y
n2 + 1
n=1
converges, that is, partial products
n2
Yn
k=1
=
2 5 10 17
· ·
·
·...
1 4 9 16
k2 + 1
converge to a finite limit as n → ∞.
k2