Lecture 38 - 3/23
Fun fact: Define the function µ(n) to be equal to 1 if n is a product of an even number of
distinct primes, −1 if n is equal to a product of an odd number of distinct primes, and 0 if n is
divisible by the square of sum prime. For example, µ(1) = 1, µ(2) = −1, µ(3) = −1, µ(4) = 0,
µ(5) = −1, µ(6) = 1, µ(7) = −1, µ(8) = 0, µ(9) = 0 and µ(10) = 1.
It is unknown for which values of s the sum
∞
X
µ(n)
ns
n=1
converges. Mathematicians can only prove it converges for s ≥ 1, but believe it converges for
any s > 1/2. If you can prove it converges for any s > 1/2, you will win a prize of $1000000.
Worksheet on Section 11.5
Instructions:
• Pair up with one or two (preferably two) other people in the class (at least one of whom
you don’t know).
• Pick one of the problems below to work on. (You should all work on the same problem).
• Read the problem and discuss possible approaches.
• Agree on an approach as a group. [If you have trouble, ask me.]
• Carry out the approach individually - if you get stuck, ask another member of your
group. [If you don’t get sufficient enlightenment from within you group, ask me.]
• When all the members in your group have finished, check your answers to see if they
agree, and if they are reasonable. Also, check with me to see if your answer is correct.
I strongly recommend completing all of the problems on this worksheet. At the
very least, please read the solutions I write.
1. What does the alternating series test state? What are the three conditions on the sequence
that must be satisfied? What happens if the third condition isn’t satisfied?
2. Determine if the following series converge.
P
(−1)n−1
(a) ∞
n=1 ln(1+n)
P
(−1)n−1
(b) ∞
n=1 ln(2+1/n) .
P
n cos(nπ)
(c) ∞
.
n=1
2n
P∞
(d) n=1 (−1)n cos πn .
√
P
√
n
(e) ∞
n).
n=1 (−1) ( n + 1 −
1
2
3. Suppose that {bn }P
is a sequence of numbers so that (i) bn > 0 for all n, and (ii) bn+1 ≥ bn
n−1
for all n. Prove that ∞
bn diverges.
n=1 (−1)
4. The alternating series test implies that
and let sn =
1
ln(2)
−
1
ln(3)
+
1
ln(4)
+ ··· +
(−1)n
n=2 ln(n)
P∞
(−1)n−1
ln(n)
converges. Let s be the value of the sum,
be the nth partial sum.
How large does n have to be to ensure that |s − sn | < 1/100?
5. Suppose that
limn→∞ bn = 0.
P∞
n=1 (−1)
n−1
bn is a series with bn > 0 for all n, bn+1 < bn for all n, and
Let sn = b1 − b2 + b3 − b4 + · · · + (−1)n−1 bn be the nth partial sum. Prove that
diverges.
P∞
n−1
sn
n=1 (−1)