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Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Chapter 2: Sequences and Series
Peter W. [email protected]
Initial development byKeith E. Emmert
Department of MathematicsTarleton State University
Fall 2011 / Real Anaylsis I
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 1Associativity need not hold when dealing with infiniteseries. Suppose that
S = 1− 12+
13− 1
4+
15− · · · =
∞∑n=1
(−1)n+1 1n.
Then we can add half the sum to the original sum andobtain
12S = 1
2 −14 +1
6 −18 + · · ·
+S = 1 −12 +1
3 −14 +1
5 −16 +1
7 −18 − · · ·
32S = 1 +1
3 −12 +1
5 +17 −1
4 + · · ·
Note that we have rearranged the original sum andobtained a completely different number!
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 2Another example:
(−1 + 1) + (−1 + 1) + · · · = 0 + 0 + · · · = 0
and moving parenthesis one step over,
−1 + (1− 1) + (1− 1) + · · · = −1 + 0 + 0 + · · · = −1.
Remark 3The conclusion: manipulations that are legal in the finiteworld need not extend to the infinite world...and we haveyet to do something really creepy, like multiplying twoinfinite series!
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Definitions and Examples
Definition 4A sequence is a function whose domain is N.
Example 5The following are various ways to represent the samesequence:
I(1, 1
2 ,13 , · · ·
)I{1
n
}∞n=1
I{1
n
}n∈N
I{1
n
}n≥1
I (an) where an = 1n for each n ∈ N.
Remark 6Sometimes our sequence will start at n0 which may ormay not equal 1.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Definitions
Definition 7 (Convergence of a Sequence)A sequence (an) converges to a real number a if, forevery positive number ε, there exists an N ∈ N such thatwhenever n ≥ N it follows that |an − a| < ε.
Some common notations for convergence are
limn→∞
an = a,
an → a as n→∞,or
an −→n→∞a.
Remark 8Note that N depends on the choice of ε!!!!!
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Definitions
Definition 9Given a real number a ∈ R and a positive number ε > 0,the set
Vε(a) = {x ∈ R | |x − a| < ε}
is called the ε−neighborhood of a.
Remark 10Note that Vε(a) = (a− ε,a + ε) is nothing more than anopen interval with a number, a, as its center.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
An Equivalent Definition of Convergence
Definition 11 (Convergence of a Sequence:Topological Version)A sequence (an) converges to a if, given anyε−neighborhood Vε(a) of a, there exists a point in thesequence after which all of the terms are in Vε(a). Inother words, every ε−neighborhood contains all but afinite number of the terms of (an).
Remark 12The idea here is that there exists an N ∈ N such thatan ∈ Vε(a) for all n ≥ N.
Remark 13Note that N depends on the choice of ε!!!!!
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
The Game of Convergence
Remark 14I We are given an ε > 0 by our worst enemy.I List the rule for choosing N ∈ N. (This requires much
scratch work.)I Demonstrate that your choice for N works by
assuming n ≥ N and showing that |an − a| < ε.
Remark 15Thus, all convergence proofs will begin as follows:
Let ε > 0 be arbitrary. Choose a naturalnumber N satisfying
N > some function of ε.
Let n ≥ N. Then,...
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 16Prove that lim
n→∞
2n5n − 3
=25.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Divergence of Sequences
Definition 17A sequence diverges if it does not converge.
Remark 18I In section 2.5 we’ll find a good way to show
divergence.I But, using the definition to disprove convergence, you
would pick a particular ε and show that no N works.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 43 – 44Problem: 2.2.1, 2.2.2, 2.2.5, 2.2.8
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Bounded Sequences
Definition 19A sequence (xn) is bounded if there exists a numberM > 0 such that |xn| ≤ M for all n ∈ N.
Theorem 20Every convergent sequence is bounded.Proof:
I
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Some Limit Laws
Theorem 21 (Algebraic Limit Theorem)Let lim
n→∞an = a and lim
n→∞bn = b, then
1. limn→∞
(can) = ca, for all c ∈ R.
2. limn→∞
(an + bn) = a + b.
3. limn→∞
(anbn) = ab.
4. limn→∞
an
bn=
ab
, provided b 6= 0.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Comparing Limits
Theorem 22 (Order Limit Theorem)Let lim
n→∞an = a and lim
n→∞bn = b.
1. If an ≥ 0 for all n ∈ N, then a ≥ 0.2. If an ≤ bn for all n ∈ N, then a ≤ b.3. If there exists c ∈ R for which c ≤ bn for all n ∈ N,
then c ≤ b. Similarly, if an ≤ c for all n ∈ N, thena ≤ c.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 49 – 50Problems: 2.3.2, 2.3.3, 2.3.5, 2.3.7, 2.3.8, 2.3.10
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
The Monotone Convergence Theorem, MCT
Definition 23I A sequence (an) is increasing if an ≤ an+1 for all
n ∈ N.I A sequence (an) is decreasing if an ≥ an+1 for all
n ∈ N.I A sequence is monotone if it is either increasing or
decreasing.
Theorem 24 (Monotone Convergence Theorem)If a sequence is monotone and bounded, then itconverges.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Sequences
Definition 25Let (bn) be a sequence. An infinite series is a formalexpression of the form
∞∑n=1
bn = b1 + b2 + b3 + · · · .
We define the corresponding sequence of partial sums(sM) by
sm = b1 + b2 + · · ·+ bm,
and say that the series∞∑
n=1
bn converges to B if the
sequence (sm) converges to B. In this case we write∞∑
n=1
bn = B.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 26
Show that the sequence∞∑
k=1
1k· 1
2k converges.
Example 27
Show that the Harmonic series∞∑
n=1
1n
diverges.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Cauchy Condensation Test
Theorem 28 (Cauchy Condensation Test)Suppose (bn) is decreasing and satisfies bn ≥ 0 for all
n ∈ N. Then, the series∞∑
n=1
bn converges if and only if the
series∞∑
n=0
2nb2n = b1 + 2b2 + 4b4 + 8b8 + · · ·
converges.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
p-Series Convergence
Corollary 29
The series∞∑
n=1
1np converges if and only if p > 1.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 54 – 55Problems: 2.4.2, 2.4.5, 2.4.6
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Subsequences
Definition 30Let (an) be a sequence of real numbers, and letn1 < n2 < n3 < · · · be an increasing sequence of naturalnumbers. Then the sequence
an1 ,an2 ,an3 , · · ·
is called a subsequence of (an) and is denoted by (anj ),where j ∈ N indexes the subsequence.
Theorem 31Subsequences of a convergent sequence converge to thesame limit as the original sequence.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 32Let 0 < b < 1. Prove that lim
n→∞bn = 0.
Example 33Prove that the sequence (−1)n does not converge.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Bolzano–Weirstrass Theorem
Theorem 34 (Bolzano-Weirstrass Theorem)Every bounded sequence contains a convergentsubsequence.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Caution!
Remark 35I The Bolzano–Weirstrass Theorem simply states that
bounded sequences have convergentsubsequences.
I It does NOT state that the original sequenceconverges!
I (−1)n is bounded, has two convergentsubsequences, but it does not converge.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 57 – 58Problems: 2.5.2, 2.5.3, 2.5.4, 2.5.6
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Cauchy Criterion
Remark 36Recall: A sequence (an) converges to a real number a if,for every ε > 0, there exists an N ∈ N such that whenevern ≥ N it follows that |an − a| < ε.
Definition 37A sequence (an) is called a Cauchy sequence if, forevery ε > 0, there exists an N ∈ N such that wheneverm,n ≥ N it follows that |an − a| < ε.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Some Theory
Theorem 38Every convergent sequence is a Cauchy sequence.Proof:
Lemma 39Cauchy sequences are bounded.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
The Cauchy Criterion
Theorem 40 (Cauchy Criterion)A sequence converges if and only if it is a Cauchysequence.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 61 – 62Problems: 2.6.1, 2.6.4, 2.6.6
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Series Theory
Remark 41I This next theorem states that the distributive
property holds and that we can add two series.I However, nothing is said about commutativity of
series or the products of series!
Theorem 42 (Algebraic Limit Theorem for Series)
If∞∑
k=1
ak = A and∞∑
k=1
bk = B, then
1.∞∑
k=1
cak = cA for all c ∈ R
2.∞∑
k=1
(ak + bk ) = A + B.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Cauchy Criterion for Series
Theorem 43 (Cauchy Criterion for Series)
The series∞∑
k=1
ak converges if and only if, given ε > 0,
there exists an N ∈ N such that whenever n > m ≥ N itfollows that
|am+1 + am+2 + · · ·+ an| < ε.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Way Cool Tests
Remark 44I Remember that when dealing with infinite series, the
first n terms are unimportant...it only matters whathappens to the (infinite) tail of the sequence/series.
I Hence, theorems can be modified to start at the nth
term rather than the first term.
Theorem 45 (nth Term Test)
If the series∞∑
k=1
ak converges, then (ak ) −−−→k→∞
0.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Way Cool Tests
Theorem 46 (Comparison Test)Assume (ak ) and (bk ) are sequences satisfying0 ≤ ak ≤ bk for all k ∈ N.
1. If∞∑
k=1
bk converges, then∞∑
k=1
ak converges.
2. If∞∑
k=1
ak diverges, then∞∑
k=1
bk diverges.
Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Geometric Series
Example 47
Suppose a 6= 0. The series∞∑
k=1
ar k converges if and only
if |r | < 1.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Way Cool Tests
Theorem 48 (Absolute Convergence Test)
If the series∞∑
k=1
|ak | converges, then∞∑
k=1
ak converges as
well.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Way Cool Tests
Theorem 49 (Alternating Series Test)Let (an) be a sequence satisfying
I a1 ≥ a2 ≥ a3 ≥ · · · ≥ an ≥ an+1 ≥ · · · andI (an) −−−→n→∞
.
Then, the alternating series∞∑
n=1
(−1)n+1an converges.
Definition 50
I If∞∑
n=1
|an| converges, then the series∞∑
n=1
an
converges absolutely.
I If∞∑
n=1
an converges, but∞∑
n=1
|an| diverges, then the
series∞∑
n=1
an converges conditionally.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Example
Example 51
Classify the converges of the series∞∑
n=1
(−1)n+1
n.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Rearrangements
Definition 52
Let∞∑
n=1
an be a series. A series∞∑
n=1
bn is called a
rearrangement of∞∑
n=1
an if there exists a one-to-one,
onto function f : N→ N such that bf (k) = ak for all k ∈ N.
Theorem 53
If∞∑
n=1
an converges absolutely, then any rearrangement of
this series converges to the same limit.Proof:
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Homework
Pages: 67 – 69Problems: 2.7.4, 2.7.6, 2.7.9, 2.7.12, 2.7.13, 2.7.14
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Double Sums & Products
Read Me.
Chapter 2:Sequences and
Series
PWhite
Discussion
The Limit of aSequence
The Algebraic andOrder LimitTheorems
MCT & InfiniteSeries
Bolzano-Weierstrass
The CauchyCriterion
Properties ofInfinite Series
Double Sums &Products
Epilogue
Overview
Discussion: Rearrangements of Infinite Series
The Limit of a Sequence
The Algebraic and Order Limit Theorems
The Monotone Convergence Theorem and a First Look atInfinite Series
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion
Properties of Infinite Series
Double Summations and Products of Infinite Series
Epilogue