Lecture 36 - 3/18
Fun fact: No one knows if the series
∞
X
n=1
n3
1
sin2 (n)
converges or not. (This series is called the “Flint Hills Series”.)
The denominator can never be zero, because sin(x) is only zero when x is an integer multiple
of π, and if n = kπ for some integers k and n, then π = n/k is rational (but we know it’s
not). Whether or not this series converges depends on how accurately π can be approximated
by rational numbers. There are some good reasons to guess the series does converge, but no
one really knows.
Worksheet on Section 11.4
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. Mark each statement as always true, sometimes true, or always false. Give examples to
support your answer.
P∞
P∞
(a)
Suppose
that
a
is
a
series
with
positive
terms,
a
≤
b
and
n
n
n
n=1 bn diverges. Then
n=1
P∞
a
converges.
n=1 n
Circle one: Always true
Sometimes true
Always false
P∞
P∞
(b)
P∞Suppose that n=1 an is a series with positive terms, an ≥ bn and n=1 bn diverges. Then
n=1 an converges.
Circle one: Always true
Sometimes true
Always false
P∞
P∞
(c)
Suppose
that
a
is
a
series
with
positive
terms,
a
≤
b
and
n
n
n
n=1
n=1 bn converges. Then
P∞
a
converges.
n
n=1
Circle one: Always true
Sometimes true
Always false
1
2
P∞
P∞
(d)
Suppose
that
a
is
a
series
with
positive
terms,
a
≥
b
and
n
n
n
n=1
n=1 bn converges. Then
P∞
n=1 an converges.
Circle one: Always true
Sometimes true
Always false
2. Use the comparison or limit comparison test to determine if the series below converge:
P
100n
(a) ∞
n=2 n2 −1 .
P
e1/n
(b) ∞
n=1 n2 .
P
1
(c) ∞
n=1 9n −8n .
P
1/n
(d) ∞
− 1.
n=1 n
3. Use any of the techniques we have discussed so far to determine if the series below converge.
P
1
(a) ∞
n=1 nn .
P
1 n
(b) ∞
.
n=1 1 + n
P∞
(c) n=2 √ 1 − √ 1
.
ln(n)
(d)
ln(n+1)
P∞
1
n=1 ne .
P∞
n=1 1 −
cos(1/n).
P
4. Suppose that ∞
n=1 an is a convergent series with positive terms.
(e)
(a) Explain why limn→∞ an = 0.
P
(b) Prove that ∞
n=1 ln(1 + an ) converges too.