Mth 311 Exam F97 Name:
Bent Petersen 311f97ex.tex December 10, 1997 Time: 110 minutes.
Instructions: =⇒
If you do not read the
instructions, then how
will you know what to
do? Read them now.
Be sure to write
your name in the
space above.
•You may use one note–sheet prepared in advance. You must put your name on your note–
sheet. Your note–sheet may not be larger than 8.5 by 11 inches (22 by 28 cm). You may
write on both sides of your note–sheet.
•You may not use any books nor additional note sheets.
•You may use a calculator, but it is difficult to imagine what you will use it for. Calculators
and other equipment may not be shared.
•Write your solutions neatly. Use the backs of the examination sheets for scratch work. There
is a blank (mostly) page at the end of the test which may also be used for rough work.
•Partial credit will be assigned only for clear, correct, legible work. Remember the emphasis
is on clear correct mathematical arguments. If you tell me that
∞
X
n=1
1
n2
converges, you are right, but you will not get much of a grade unless you also tel l me con-
vincingly how you deduced it converges.
Problem 1. (40 points).
Part (AA): If a≥0andb≥0compute√b−√a2to obtain an upper bound for 2√ab.
Part (BB): If an≥0, bn≥0foreachn∈
N
and the series P∞
n=1 anand P∞
n=1 bnconverge
give a careful proof that P∞
n=1 √anbnconverges.
