Lecture 13 Eat the frog - Brunel University London

Overview (MA2730,2812,2815) lecture 13

Lecture slides for MA2730 Analysis I

Simon Shawpeople.brunel.ac.uk/~icsrsss

[email protected]

College of Engineering, Design and Physical Sciencesbicom & Materials and Manufacturing Research InstituteBrunel University

October 26, 2015

Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel

MA2730, Analysis I, 2015-16


Contents of the teaching and assessment blocks

MA2730: Analysis I

Analysis — taming infinity

Maclaurin and Taylor series.

Sequences.

Improper Integrals.

Series.

Convergence.

LATEX2ε assignment in December.

Question(s) in January class test.

Question(s) in end of year exam.

Web Page:http://people.brunel.ac.uk/~icsrsss/teaching/ma2730




Lecture 13

MA2730: topics for Lecture 13

Lecture 13

A criteria for divergence

Telescopic series, harmonic series

Algebra of series

Examples and Exercises

Reference: The Handbook, Chapter 4, Section 4.2.Homework: Questions 1, 4, 8 on Sheet 3aSeminar:

∑k−2 proof. Questions 4, 8 on Sheet 3a.




Lecture 13

Study Habits, and Getting Things Done

Time Management — Tip 1

Eat the frog

Eat a live frog first thing in the morning and nothing worse willhappen to you for the rest of the day.Mark Twain




Lecture 13

Reference: Stewart, Chapter 12.2.We have seen that:

A series is obtained by adding all the terms of a sequence.

A series converges if the sequence of partial sums converges:

If sn → s where sn =n∑

k=1

ak then s =∞∑

k=1

ak.

If |r| < 1 the geometric series converges:

∞∑

k=0

ark =a

1− r.

Alternating sign series can be **** pathological ****.

Today we continue our study of series.




Lecture 13

Given a series the two important questions are:

1 Does the sum exist?

2 What is its value?

The first question takes priority — if the answer is NO then thesecond question is irrelevant.

From Subsection 4.2.1, A Criterion for Divergence, we have thefollowing useful test.

Theorem 4.7, Divergence Criterion

If∞∑

k=1

ak is convergent, then limk→∞

ak = 0.

Let’s unpick this.




Lecture 13


If∞∑

k=1

ak is convergent, then limn→∞

an = 0.

Another way of writing this is:


If∑∞

1 ak exists then ak → 0 as k → ∞.

Which do you prefer?Which would you write?There’s no right way.But there are plenty of wrong ways!We need a proof. . .




Lecture 13


If∑∞


Proof

We know that sn → s, where sn =∑n

1 ak, and by ‘adding zero’ weget,

an = sn − sn−1 = (sn − s) + (s− sn−1).

By the algebra of limits we then see that,

limn→∞

an = limn→∞

(sn − s) + limn→∞

(s− sn−1) = 0.

Oddly perhaps, adding zero, as in the red line above, can beuseful. . .




Lecture 13


If∑∞


This says:

If the series converges then the terms in the tail of the sum musttend to zero.

It does not say:

If the terms in the tail of the sum tend to zero then the series mustconverge.

Be careful!This inversion is wrong but can form a subconscious trap.




Lecture 13

Inversion

Parent says to child:

If you don’t clean your room you wont get any ice cream.

Child cleans room (gasp!)

Child says to parent:

I’ve cleaned my room so I have now come for ice cream.

Is the child entitled to ice cream?

Borrowed from: Chapter 8, How to Think Like a Mathematician, Kevin

Houston, CUP 2009.




Lecture 13

Inversion

This subconscious inversion can be a trap door through which eventhe very experienced can fall.

For example: x = 3 so x2 = 9.

Inverts to: x2 = 9 so x = 3.Comments?That was easy to spot, but are these equivalent?

If it’s sunny, I’ll walk the dog

If I’m walking the dog, then it’s sunny




Lecture 13

So, to summarise


If∑∞


It does say:

If the series converges then the terms in the tail of the sum musttend to zero.

It does not say:

If the terms in the tail of the sum tend to zero then the series mustconverge.

Be careful!




Lecture 13

Examples: which diverge?

∞∑

k=1

sin(k) sin(k) 6→ 0 DIVERGENT

∞∑

k=1

1

k21

k2→ 0 ?

∞∑

k=1

k + 1

k + 2

k + 1

k + 2=

1 + 1/k

1 + 2/k→ 1 DIVERGENT

∞∑

k=1

1

k

1

k→ 0 ?




Lecture 13

Telescopic series

Here’s something new. Determine the sum, if it exists:∞∑

k=1

2

(k + 2)(k + 4)

Recall partial fractions from last year:

2

(k + 2)(k + 4)=

1

k + 2− 1

k + 4

Hence:∞∑

k=1

2

(k + 2)(k + 4)=

∞∑

k=1

(1

k + 2− 1

k + 4

)=

7

12.

BoardworkShaw bicom, mathematics, CEDPS, IMM, CI, Brunel



Lecture 13

Exercise

Given the partial fraction decomposition

8

(5− k)(k + 3)=

1

k + 3+

1

5− k

Determine ∞∑

k=6

8

(5− k)(k + 3)

Boardwork




Lecture 13

Telescoping Series

The main point is that all except a finite number of terms in thesum repeat, but with a different sign.

They therefore cancel each other out.

Definition 4.8 in Subsection 4.2.2, Telescopic Series

A finite sum sn =∑n

k=1 ak in which subsequent terms cancel eachother, leaving only the initial and final terms, is called a telescopicsum.If the final terms tend to zero with n the telescopic series,

∑∞1 ak,

is determined by just the initial terms.

It’s good when it happens, but not all series telescope in this way.




Lecture 13

A review of partial fraction (PF) decomposition

factor in term in PF

denominator decomposition

ax+ bA

ax+ b

(ax+ b)2A

ax+ b+

B

(ax+ b)2

ax2 + bx+ cAx+ b

ax2 + bx+ c

(ax2 + bx+ c)2Ax+ b

ax2 + bx+ c+

Cx+D

(ax2 + bx+ c)2




Lecture 13

Something new: The Harmonic Series

Definition: The Harmonic Series

The infinite sum,

∞∑

n=1

1

n= 1 +

1

2+

1

3+

1

4+

1

5+

1

6+ . . .

is called The Harmonic Series.

Theorem 4.9, Subsection 4.2.3

The Harmonic Series diverges.

It is the canonical example a divergent series,∑∞

1 ak, for whichthe terms ak → 0.




Lecture 13

Before proving this we need a lemma

Lemma

1 +1

2+

N∑

k=2

2k∑

n=2(k−1)+1

1

n> 1 +

N

2.

Proof

This is a rough ride.We’ll look at the main steps.You should study it.We’ll look at a similar proof in the seminar.




Lecture 13

Proof

We expand the outer sum and take lower bounds:

1 +1

2+

N∑

k=2

2k∑

n=2(k−1)+1

1

n

= 1 +1

2+

(1

3+

1

4

)+

(1

5+ · · ·+ 1

8

)+ · · ·

· · ·+(

1

2(N−1) + 1+ · · ·+ 1

2N

)

> 1 +1

2+

2

22+

22

23+

23

24+ · · ·+ 2(N−1)

2N

= 1 +N

2.




Lecture 13

Theorem 4.9, Subsection 4.2.3

The Harmonic Series diverges.

Proof

The idea is to sum over the lengths between powers of two and usethe lemma:

∞∑

n=1

1

n= 1 +

1

2+ lim

N→∞

N∑

k=2

2k∑

n=2(k−1)+1

1

n> lim

N→∞

(1 +

N

2

).

Therefore ∞∑

n=1

1

n> 1 + lim

N→∞N

2= ∞

and we conclude that the harmonic series is divergent.




Lecture 13

Seminar: masterclass

That was difficult!It will be useful to see more like that.So. . . Half of next week’s seminar will be devoted to a proof of thefollowing theorem.

Theorem (Lemma 4.11 in The handbook)

The sum of the squares of the reciprocals of all the naturalnumbers exists and is bounded by 67/36.

Theorem (alternative)

∞∑

k=1

1

k26 67

36.

You can think of that part of the seminar as a masterclass.Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel



Lecture 13

Algebra of Limits for Series, Theorem 4.13 in Subsection 4.2.4

Let c ∈ R be a constant, let∑∞

k=1 ak converge to a and∑∞

k=1 bkconverge to b. Then,

1

∞∑

k=1

(ak + bk) converges to a+ b;

2

∞∑

k=1

(ak − bk) converges to a− b;

3

∞∑

k=1

cak converges to ca.

We use this exactly as we used the algebra of limits for sequences:to break big problems up into smaller ones; address each smallerone in isolation; and, then assemble the results to solve the biggerproblem.




Lecture 13

Summary

We can

Use ak 6→ 0 as k → ∞ to recognise a divergent series.

Use partial fractions to investigate whether or not a sumtelescopes and, if it does, to investigate the resulting sum ofthe series.

Recognise the harmonic series and prove that it diverges.




End of Lecture

Computational andαpplie∂ Mathematics

Eat the frog

Eat a live frog first thing in the morning and nothing worse willhappen to you the rest of the day.Mark Twain

Reference: The Handbook, Chapter 4, Section 4.2.Homework: Questions 1, 4, 8 on Sheet 3aSeminar:

∑k−2 proof. Questions 4, 8 on Sheet 3a.



Lecture 13 Eat the frog - Brunel University London

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Transcript of Lecture 13 Eat the frog - Brunel University London