Week 9 recitation questions
A. Preamble
In lectures, we defined sequences and series, and their convergence. A sequence is an ordered list {an } =
{a1 , a2 , a3 , . . .} with a first element but no last element. We discussed a few examples:
1 3 7
1
1− n =
, , , . . . , {(−1)n log(n)} = {− log(1), log(2), − log(3), . . .} , {1, 1, 2, 3, 5, 8, . . .} .
2
2 4 8
The sequence {an } converges to L if lim an = L.
n→∞
1. Determine whether the following sequences converge or diverge; and if they converge, what they converge
to.
(a) 1 − 21n .
(b) {(−1)n log(n)}.
n
o
(c) (−1)n sin(n)
.
n
A series is a formal infinite sum of terms:
X
an =
n≥1
The series
X
an = a1 +a2 +a3 +· · · . We discussed a few examples:
n=1
n≥1
X
∞
X
1
,
2n−1
X
n≥1
n,
X1
.
n
n≥1
an converges to L if its sequence of partial sums {Pn } = {a1 , a1 + a2 , a1 + a2 + a3 , . . .} con-
n≥1
verges to L.
2. Determine what
X
n≥1
1
converges to.
2n−1
We generalized the answer to this question and proved the following result: the geometric series
X
arn−1
n≥1
(where a, r 6= 0) converges to
a
1−r
if |r| < 1, and diverges otherwise.
It turns out that convergence of the individual terms
X in a series to 0 is a necessary but not sufficient condition
for convergence. This is the Divergence Test: if
an converges, then lim an = 0. (Note that the converse
n→0
n≥1
is not true: if lim an = 0, it need not be true that
n→0
3. Determine if
X
an converges, as the following question demonstrates.)
n≥1
X1
converges.
n
n≥1
Z
This question was answered by comparing the series to the divergent improper integral
1
comparison can be formalized into the Integral Test.
1
∞
1
dx. This
x
B. Questions
Z
∞
1
dx diverges.
x
1
X1
(b) Prove that
diverges.
n
1. (a) Prove that
n≥1
2. (a) Prove the Integral Test: if an = f (n), where f (x) is continuous, positive, and non-increasing for
x ≥ 1, then we have the following.
Z ∞
X
f (x)dx converges, then
an converges.
i. If
1
Z
n≥1
∞
f (x)dx diverges, then
ii. If
1
X
an diverges.
n≥1
(b) Prove that
X 1
diverges if p ≤ 1 and converges if p > 1.
np
n≥1
X n3
converges.
en4
n≥1
X
1
(b) Determine if
converges.
n log(n)
n≥1
X n
(c) Determine if
converges.
n2 + 5
3. (a) Determine if
n≥1
1
converges.
n(n + 1)
n≥1
X
1
(b) Determine what
converges to.
n(n + 1)
n≥1
X
5. Prove that
1.21/n diverges.
4. (a) Prove that
X
n≥1
6. Prove that if
X
n≥1
7. Prove that
X
n≥1
an and
X
bn are convergent series with all positive terms, than
n≥1
an converges if, and only if,
X
an bn converges.
n≥1
X
an converges for any positive integer k.
n≥k
2