Lecture 20people.brunel.ac.uk/~icsrsss/teaching/ma2730/lec/print8... · 2015. 11. 20. · Maclaurin...

Overview (MA2730,2812,2815) lecture 20

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

November 20, 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 20

MA2730: topics for Lecture 20

Lecture 20

Power series

Radius of Convergence

The natural domain of a power series function rule

Term-by-term differentiation of Taylor series

Examples and Exercises

Reference: The Handbook, Chapter 7, Section 7.2.Homework: Sheet 6a, Questions 1 and 3Seminar: Ad Hoc. Student driven




Lecture 20

Clarity, Brevity and Accuracy

The Assignment — a comment on page limits

I didn’t have time to write a short letter, so I wrote a long oneinsteadMark Twain




Lecture 20

Reference: Stewart, Chapter 12.

We’ve come a long way.

Improper integrals, Taylor and Maclaurin series. . . .

. . . sequences, series, convergence, limits.

We have seen that Maclaurin series of functions sometimeswork. . .

. . . and sometimes don’t

For example in the interval |x| < 1 we found that,

1

1− x= 1 + x+ x2 + x3 + x4 + · · · (if and only if −1 < x < 1).

Today we study this interval of convergence in more general terms:it is called radius of convergence.




Lecture 20

This is the fundamental concept.

Power Series, Definition 7.6, the Handbook

A power series is a series of the form

∞∑

n=0

cn(x− a)n,

where {cn}∞n=0 is a sequence of real numbers.

The case a 6= 0 reminds us of Taylor series and the case a = 0, ofMaclaurin series.

For clarity, we will often focus mainly on the case a = 0.




Lecture 20

Here it is again.

Power Series, Definition 7.6, the Handbook

A power series is a series of the form

∞∑

n=0

cn(x− a)n,

where {cn}∞n=0 is a sequence of real numbers.

In a power series x plays its usual role of an independent variable.So we can interpret a power series as a function rule.

f(x) =

∞∑

n=0

cn(x− a)n.

Then f : D → R where D is the natural domain for the function.

How could we define D?Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel



Lecture 20

The main mathematical notions and tools are in The Handook:

Definition 7.8 — based on the ratio and root tests

Lemma 7.9 — the root and ratio tests are ‘equal’

Theorem 7.10 — the main result

Let’s review these.




Lecture 20

Radius of Convergence. Definition 7.8

Let∞∑

n=0

cn(x− a)n be a power series and let

ρ1 = limn→∞

∣∣∣∣cn+1

cn

∣∣∣∣ and ρ2 = limn→∞

|cn|1/n.

The radius of convergence R of the series is equal to

1 R = 1/ρ2 when 0 < ρ2 < ∞,

2 R = 0 when ρ2 = ∞,

3 R = ∞ when ρ2 = 0.




Lecture 20

Lemma 7.9

In Definition 7.8, when ρ1 and ρ2 exist, they are equal.More precisely: ρ2 = ρ1 whenever ρ1 exists.

Remark

The subtlety is that ρ2 might exist when ρ1 doesn’t.

Outline proof

For a positive sequence, the main observation is that ifan+1/an → L > 0 then, given ǫ > 0, there is an N ∈ N such thatfor n > m > N ,

∣∣∣∣lnan+1

an− lnL

∣∣∣∣ < ǫ =⇒ (n−m) lnL ≈n∑

k=m

lnak+1

ak.

The details are quite technical — PROOF IS NOT EXAMINABLE.




Lecture 20

Pointwise convergence of a power series, Theorem 7.10

Given a power series∞∑

n=0

cn(x− a)n, either:

1 the series converges for all x when R = ∞;

2 the series converges just for x = a when R = 0;

3 the series converges for all x with |x− a| < R.

In the last case we need to check separately what happens at theboundary points where |x− a| = R. These are the points

x = a+R and x = a−R.

Proof

Boardwork (use the Cauchy root test).




Lecture 20

Example 1

Determine the radius of convergence, R, for the power series

f(x) =

∞∑

n=1

(x− 1)n

n 2n

and investigate its behaviour at the boundary |x− a| = R.

Solution: Boardwork




Lecture 20

It is important to check the boundary convergence behaviourseparately because anything can happen!From Subsection 7.2.1 of The Handbook we have the followinggeneral set up. Suppose that

f(x) =

∞∑

n=0

cn(x− a)n

has radius of convergence R < ∞. Then the series converges ifa−R < x < a+R and the boundary points x = a−R andx = a+R have to investigated separately.

xaa−R a+R

RR




Lecture 20

We have the following general set up. The power series

f(x) =∞∑

n=0

cn(x− a)n

has radius of convergence R < ∞ and so converges in |x− a| < R.

At the left boundary point, x = a−R:

f(x) =

∞∑

n=1

cn(−R)n.

At the right boundary point, x = a+R:

f(x) =∞∑

n=1

cnRn.

We can expect the theory of alternating series to be important.Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel



Lecture 20

Example 2

Find the natural domain of the function rule given by the powerseries

f(x) =∞∑

n=0

cn(x− a)n for cn =1

np2nand a = 1.

Boardwork

Determine R using the ratio test.

Put x = 1−R into the power series: is it convergent?

Put x = 1 +R into the power series: is it convergent?

Hence the natural domain is . . . ?




Lecture 20

The main point

We have seen that provided we remain inside the radius ofconvergence and pay careful attention to the behaviour on theboundary, a power series defines a function, f : D → R.

Several questions naturally arise.

Can we add, subtract, multiply such power series?

If we differentiate the power series do we get f ′(x)?

Can we integrate the power series and get∫f(x) dx?

The answer is yes — if we remain inside the natural domain of f .

Addition, subtraction and multiplication work in the usual way. Wewill look at differentiation From Subsection 7.2.2 of TheHandbook.




Lecture 20

Lemma 7.11, Subsection 7.2.2 in The Handbook

Let f, g : I → R be an infinitely differentiable functions with Taylor seriesT a∞f(x) =

∑∞n=0 cn(x− a)n about a ∈ I, and similarly for g.

1 The sum and product of the Taylor series of f and g about x = aare the Taylor series of the sum and product functions.

2 For k ∈ N, the Taylor series T a∞(x − a)kf(x) is given by

(x − a)kT a∞f(x) =

∞∑

n=0

cn(x− a)n+k.

3 Let b ∈ R, k ∈ N, the Maclaurin series for g(x) = f(bxk) is

T∞g(x) =

∞∑

n=0

(cnbn)xkn.

The most important part of this theorem is next. . .




Lecture 20

Lemma 7.11, Subsection 7.2.2 in The Handbook

Let f : I → R be an infinitely differentiable function with Taylorseries T a

∞f(x) =∑∞

n=0 cn(x− a)n about a ∈ I.

1 The Taylor series of the derivative f ′ of f is given by theseries of term-by-term derivatives of the Taylor series of f andhas the same radius of convergence.

Outline Proof —- the main steps

First: T a∞f ′(x) =

∞∑

n=0

f (n+1)(a)

n!(x− a)n =

∞∑

n=1

f (n)(a)

(n− 1)!(x− a)n−1.

Second:d

dxT a∞f(x) =

d

dx

∞∑

n=0

f (n)(a)

n!(x− a)n =

∞∑

n=1

f (n)(a)

(n− 1)!(x− a)n−1.

They are the same so (T a∞f(x))′ = T a

∞f ′(x).




Lecture 20

For the second part of this outline proof we put

f(x) =∞∑

n=0

f (n)(a)

n!(x− a)n =

∞∑

n=0

cn(x− a)n,

f ′(x) =∞∑

n=1

f (n)(a)

(n− 1)!(x− a)n−1 =

∞∑

n=1

bn(x− a)n.

To use the root test we first note that,

bn =f (n)(a)

(n− 1)!= n

f (n)(a)

n!= ncn,

and so ρ2 = 1/R is the same for both power series because,

limn→∞

|bn|1/n = limn→∞

n1/n limn→∞

|cn|1/n = limn→∞

|cn|1/n

since n1/n → 1 (using Part 3 of Proposition 2.14, Lecture 10).




Lecture 20

Summary

We will pick up on these observations in the next lecture.

For the moment we can:

Recognise a power series.

Interpret its natural domain in terms of R.

Use the root and ratio tests to determine R.

Investigate the convergence at the boundary |x− a| = R.

Find f ′(x) by term-by-term differentiation of f(x).




End of Lecture

Computational andαpplie∂ Mathematics

The Assignment — a comment on page limits

I didn’t have time to write a short letter, so I wrote a long oneinsteadMark Twain

Reference: The Handbook, Chapter 7, Section 7.2.Homework: Sheet 6a, Questions 1 and 3Seminar: Ad Hoc. Student driven



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Lecture 20people.brunel.ac.uk/~icsrsss/teaching/ma2730/lec/print8... · 2015. 11. 20. · Maclaurin...

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