Math 414
Professor Lieberman
May 2, 2003
PRACTICE FINAL EXAM
Directions: Solve five problems, including #6 or #7.
1. If the sequence (an ) satisfies the condition lim (an − an−2 ) = 0, show that lim (an −
n→∞
n→∞
an−1 )/n = 0.
2. The Fibonacci numbers Fn are defined by F1 = 1, F2 = 1, and Fn+2 = Fn + Fn+1 for n ∈ N.
Use mathematical induction to show that
n
X
Fk = Fn−2 − 1.
k=1
3. (a) Give an example of a function f : [a, b] → R which is discontinuous but whose range is
an open, bounded interval.
(b) Give an example of a function f : [a, b] → R which is discontinuous but whose range is
an closed, bounded interval.
4. If f is differentiable at some number x, show that, for any positive numbers a and b with
a < b,
a2 f (x + bh) − b2 f (x + ah) + (b2 − a2 )f (x)
.
f 0 (x) = lim
h→0
(a2 b − b2 a)h
5. Is the improper integral
∞
Z
sin(ex ) dx
0
convergent or divergent?
P
P
6. If |ak | converges and the sequence (bk ) is bounded, show that the series ak bk converges
absolutely.
7. Decide whether the series
∞
X
(−1)k
k=2
k ln k
converges or diverges.
8. TRUE or FALSE? If f : [0, ∞) → R and there is a sequence (xn ) such that lim f (xn ) = ∞,
n→∞
then lim f (x) = ∞.
x→∞
1
2
9. If a > 0 and b is any real number, show that the equation x3 + ax + b = 0 has exactly one
solution.
10. Show that f (x) = x sin x is not uniformly continuous on the interval [0, ∞).