.
1. Write the first four partial sums in the series .
s1=
s2=
s3=
s4=
2. Determine whether the sequence 
converges or diverges, and if it converges find the limit.
3. Determine whether the series 
converges or diverges.
4. Determine whether the series 
converges or diverges.
5. Determine whether the series 
converges or diverges.
6. Determine whether the series 
converges or diverges.
7. Malcolm is a Calculus student at Rice University who's disappointed
with his grade on his first exam. Malcolm says "Oh
dear. I do seem
to have scored rather less well than I would have liked. One question in
particular remains irksome still. We were to determine the behavior of
the series 
,
and I'm afraid I seem to have botched it terribly. I used the comparison
test, since checking the first several terms made it quite clear that the
terms in this series are less than those in 1/n2, and so all
that should make it converge. I was really put out when the Prof returned
the exam with no credit whatsoever on the problem and a note about using
the Test for Divergence. I suppose I see his point about the Divergence
and all, but I'm afraid I simply don't see what's wrong with my
way."
Explain clearly to Malcolm, in terms he can understand, either what's
wrong with his approach or how it can be reconciled with what his Professor
said.
8. [Finney/Thomas, 1990, p. 592] The figure below shows the first three
rows of a sequence of rows of semicircles. There are 2n semicircles
in the nth row, each of radius 1/2n. Find the sum
of the areas of all the semicircles.
9. Determine whether the series 
converges or diverges, where 
means taking the n2 root of 2.
10. Prove that 
=0.
[Hint: You might want to start out by investigating 
ak/k!.]
Extra Credit [this problem can replace your lowest-scoring problem on the exam]
a) Give an example of a series an
which is divergent, but for which (an)2
is convergent.
b) Show that whenever an
is a convergent series whose terms are positive, (an)2
is convergent also.