Overview (MA2730,2812,2815) lecture 13 Lecture slides for MA2730 Analysis I Simon Shaw people.brunel.ac.uk/~icsrsss [email protected]College of Engineering, Design and Physical Sciences bicom & Materials and Manufacturing Research Institute Brunel University October 26, 2015 Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 13 Contents of the teaching and assessment blocks MA2730: Analysis I Analysis — taming infinity Maclaurin and Taylor series. Sequences. Improper Integrals. Series. Convergence. L A T E X2 ε 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 Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 13 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 3a Seminar: ∑ k -2 proof. Questions 4, 8 on Sheet 3a. Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16 Overview (MA2730,2812,2815) lecture 13 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 will happen to you for the rest of the day. Mark Twain Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel MA2730, Analysis I, 2015-16
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Lecture 13 Eat the frog - Brunel University London
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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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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.
Shaw bicom, mathematics, CEDPS, IMM, CI, Brunel
MA2730, Analysis I, 2015-16
Overview (MA2730,2812,2815) lecture 13
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: