Chapter 2: Sequences and Series - Faculty Website Listing · Chapter 2: Sequences and Series PWhite...

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


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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!


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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!


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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!!!!!


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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!!!!!


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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,...


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Example

Example 16Prove that lim

n→∞

2n5n − 3

=25.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 43 – 44Problem: 2.2.1, 2.2.2, 2.2.5, 2.2.8


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 49 – 50Problems: 2.3.2, 2.3.3, 2.3.5, 2.3.7, 2.3.8, 2.3.10


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

p-Series Convergence

Corollary 29

The series∞∑

n=1

1np converges if and only if p > 1.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 54 – 55Problems: 2.4.2, 2.4.5, 2.4.6


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Example

Example 32Let 0 < b < 1. Prove that lim

n→∞bn = 0.

Example 33Prove that the sequence (−1)n does not converge.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Bolzano–Weirstrass Theorem

Theorem 34 (Bolzano-Weirstrass Theorem)Every bounded sequence contains a convergentsubsequence.Proof:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 57 – 58Problems: 2.5.2, 2.5.3, 2.5.4, 2.5.6


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Some Theory

Theorem 38Every convergent sequence is a Cauchy sequence.Proof:

Lemma 39Cauchy sequences are bounded.Proof:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue


Theorem 40 (Cauchy Criterion)A sequence converges if and only if it is a Cauchysequence.Proof:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 61 – 62Problems: 2.6.1, 2.6.4, 2.6.6


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Geometric Series

Example 47

Suppose a 6= 0. The series∞∑

k=1

ar k converges if and only

if |r | < 1.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Way Cool Tests

Theorem 48 (Absolute Convergence Test)

If the series∞∑

k=1

|ak | converges, then∞∑

k=1

ak converges as

well.Proof:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Example

Example 51

Classify the converges of the series∞∑

n=1

(−1)n+1

n.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



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:


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Homework

Pages: 67 – 69Problems: 2.7.4, 2.7.6, 2.7.9, 2.7.12, 2.7.13, 2.7.14


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Double Sums & Products

Read Me.


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Overview









Epilogue


Series

PWhite

Discussion




Bolzano-Weierstrass

The CauchyCriterion



Epilogue

Epilogue

Read me.

I

Chapter 2: Sequences and Series - Faculty Website Listing · Chapter 2: Sequences and Series PWhite...

Documents

Transcript of Chapter 2: Sequences and Series - Faculty Website Listing · Chapter 2: Sequences and Series PWhite...