Math 1321 (Qinghai Zhang)
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Summary for §8.1 - §8.4
Overview of infinite series
2013-JAN-11
Theorem 6 (Test for divergence).
P∞If limn→∞ an does
not exist or limn→∞ an 6= 0, then n=1 an diverges.
One major motivation: approximate the same number/function with different accuracies.
Theorem 7 (Integral test). Let f : R → R be a continuous bounded function satisfying f (n) = an . If ∃M ∈ N+ ,
s.t.
(x) > 0 and f 0 (x) < 0, then
• Question: how accurate can it be?
P∞ ∀x ∈ [M, +∞), Rf∞
n=1 an converges iff M f (x)dx is convergent.
• Answer: as accurate as you wish.
P
P
Theorem 8 (Comparison test). Two series
an ,
bn
We care about two things:
have positive terms and satisfy ∀n ∈ N+ , an ≤ bn .
P∞
P∞
(i) Does a series converge? It’s not useful if it diverges.
• If n=1 bn converges, then n=1 an converges.
P∞
P∞
(ii) If it converges, how many terms do we need to ob• If n=1 an diverges, then n=1 bn diverges.
P
tain the target accuracy?
Theorem
9 (Limit comparison test). If two series an ,
P
bn have positive terms and
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Basic Concepts
an
= c ∈ (0, ∞),
bn
(3)
Definition 1. A sequence is a function defined on the
set of all positive integers N+ . Or more simply, it is a then either both series converge or both diverge.
countable set {an }.
Theorem 10 (Alternating series test). Let
Pa sequence
{b
>
0}
be
given.
If
the
alternative
series
(−1)n−1 bn
n
Definition 2. A sequence {an } has the limit L, written
satisfies
≥ 1, bn+1 ≤ bn , and (ii) limn→∞ bn = 0,
as
P (i) ∀n
n−1
then
(−1)
bn converges.
lim an = L, or an → L as n → ∞,
(1)
n→∞
P
Definition 11. A series
an is called absolutely conP
if
vergent
if
the
series
|a
|
converges.
n
∀ > 0, ∃N, s.t. ∀n > N, |an − L| < .
(2)
If such a limit L exists, we say that {an } converges to L. Theorem 12. If a series is absolutely convergent, then
it is convergent.
Definition 3. A series
is
the
sum
of
all
terms
in
a
sean+1
Pn
Theorem 13 P
(Ratio test). Suppose lim
quence {an }, i.e. s = i=1 ai .
P n→∞ | an | = L
for the seriesP an . If L < 1, then
an converges; if
P
Pn
As a P
shorthand notation,
an := i=1 ai . For any L > 1, then
an diverges; if L = 1, no conclusion can
finite n,
an is finite if each ai (i = 1, 2, . . . , n) is finite. be drawn.
So
it
is
only
nontrivial to talk about the convergence of
P∞
a
.
n=1 n
4 Remainder estimation
P∞
n−1
Theorem 4. The geometric series
with
n=1 ar
P
a 6= 0 is convergent if |r| < 1 and divergent otherwise. Definition 14. For the series {sn = ni=1 ai }, the nth
a
.
In the former case, sn = 1−r
remainder Rn is the error of sn in approximating the
P∞ 1
limit S = limm→∞ sm ,
Theorem 5. The p-series
n=1 np is convergent if
∞
X
p > 1 and divergent if p ≤ 1.
Rn = S − sn =
si .
(4)
lim
n→∞
i=n+1
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Convergence tests
Theorem 15 (Remainder estimate for integral test).
Let f be a function that satisfies the conditions in Theorem 7, then
Z ∞
Z ∞
f (x)dx ≤ Rn ≤
f (x)dx.
(5)
(a) test for divergence,
(b) integral test,
(c) comparison test,
n+1
n
Adding sn to (5) proves Theorem 7.
(d) limit comparison test,
Theorem 16 (Alternating series estimation). For the
convergent alternating series defined in Theorem 10,
|Rn | ≤ bn+1 .
(e) alternating series test,
(f) ratio test.
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