Infinite sequence & series 1st lecture

Tutor:

Dr. Tariq Mahmood
Assistant Professor
Centre for High Energy Physics
University of the Punjab

Class:

B.Sc (Hons.)
Computational Physics

Copyright © 2013 CHEP, P. U. Lahore.

Lecture-1

Tutor’s Brief Introduction
Name:
Designation:
Qualification:

Dr. Tariq Mahmood Khan
Assistant Professor
Ph.D (BIT, Beijing, P. R. China) in
Computational Materials Physics
M.Phil (CHEP, P. U., Lahore, Pakistan)
tariq_mahmood78@hotmail.com

Email:
Publications:
More than 30 articles have been published in International SCI
journals with good impact factors (Physica B: Condensed Matter,
Materials Letter, The Journal of Physical Chemistry A,
Electrochimica Acta, Materials Research Bulletin, Journal of
Alloys and Compounds , Journal of Nanoscience and
Nanotechnology, Solid State Sciences, Materials Chemistry and
Physics , Materials Research Bulletin , Current NanoScience, Thin
Solid Films, Sains Malaysiana, Journal of Optoelectronics and
A d v a n c e
M a t e r i a l s .

Lecture-1

Course Description
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Classes:
34-36 (2 credit hours)
Total Marks: 100
Assignments:
25
Mid Term:
35
Final Term: 40

Note: Students with less than 75% attendance will not able to sit in
the exam.
 Book:

Calculus, Ninth Edition By Thomas and Finney
Thomas’ Calculus 11th Edition By Maurice D. Weir et al

 Chapter:

Chapter 8 and 11


Lecture-1

Syllabus
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Limits of Sequences of Numbers
Theorems for Calculating Limits of Sequences
Infinite Series
The Integral Test for Series of Nonnegative Terms
Comparison Tests for series of Nonnegative Terms
The Ratio and Root Tests for Series of Nonnegative Terms
Alternating Series, Absolute and Conditional Convergence
Power Series
Taylor and Maclaurin Series
Convergence of Taylor Series; Error Estimates
Application of Power Series


Lecture-1

Applications
Infinite sequences and series are important in physics
and engineering. One of the most well-known is
the Fourier series , which can mathematically define
certain signal waveforms.
In Materials Physics, infinite series are used to
calculate different calculations (Electric, mechanical,
optical, etc) in the form of wave functions.


Lecture-1

Sequences


Lecture-1

Objectives
 List the terms of a sequence.
 Determine whether a sequence converges or

diverges.
 Write a formula for the nth term of a sequence.
 Recursion formula


Lecture-1

Sequences
A sequence is defined as a function whose domain is
the set of positive integers. Although a sequence is a
function, it is common to represent sequences by
subscript notation rather than by the standard function
notation. For instance, in the sequence
Sequence

1 is mapped onto a1, 2 is mapped onto a2, and so on. The
numbers a1, a2, a3, . . ., an, . . . are the terms of the
sequence. The number an is the nth term of the
sequence, and the entire sequence is denoted by {an}.

Lecture-1

Sequences
A sequence is a function that has a set of natural
numbers as its domain.
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f (x) notation is not used for sequences.
Write an f (n)
Sequences are written as ordered lists
a1 , a2 , a3 , ...

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a1 is the first element, a2 the second element,
and so on


Lecture-1

Example 1 – Listing the Terms of a Sequence
a. The terms of the sequence {an} = {3 + (–1)n} are
3 + (–1)1, 3 + (–1)2, 3 + (–1)3, 3 + (–1)4, . . .
2,
4,
2,
4,
....
b. The terms of the sequence {bn}


are

Lecture-1

Example 1 – Listing the Terms of a Sequence
c. The terms of the sequence {cn}

cont’d

are

d. The terms of the recursively defined sequence {dn},
where d1 = 25 and dn + 1 = dn – 5, are
25, 25 – 5 = 20, 20 – 5 = 15, 15 – 5 = 10,. . . . .

Lecture-1

Sequences
A sequence is often specified by giving a formula for
the general term or nth term, an.
Example Find the first four terms for the sequence

an

n 1
n 2

Solution

a1

(1 1) /(1 2)

a3

(3 1) /(3 2)


2 / 3, a2
4 / 5, a4

(2 1) /(2 2) 3/ 4
(4 1) /(4 2) 5 / 6
Lecture-1

Graphing Sequences
The graph of a sequence, an, is the graph of the
discrete points (n, an) for n = 1, 2, 3, …
Example Graph the sequence an = 2n.
Solution


Lecture-1

Sequences
 A finite sequence has domain the finite set

{1, 2, 3, …, n} for some natural number n.
Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
 An infinite sequence has domain

{1, 2, 3, …}, the set of all natural numbers.
Example 1, 2, 4, 8, 16, 32, …


Lecture-1

Convergent and Divergent Sequences
 A convergent sequence is one whose terms get
closer and closer to a some real number. The
sequence is said to converge to that number.
 A sequence that is not convergent is said to be
divergent.


Lecture-1

Example The sequencean

1
n

converges to 0.

The terms of the sequence 1, 0.5, 0.33.., 0.25, …
grow smaller and smaller approaching 0. This can
be
seen graphically.


Lecture-1

2

Example The sequence an n is divergent.
The terms grow large without bound
1, 4, 9, 16, 25, 36, 49, 64, …
and do not approach any one number.


Lecture-1

Sequences and Recursion Formulas
 A recursion formula or recursive definition

defines a sequence by
 Specifying the first few terms of the sequence
 Using a formula to specify subsequent terms in terms of

preceding terms.
OR
 a n term can be calculated directly from the value of n.
But sequences are defined recursively by giving
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The value(s) of the initial term(s)
The rule, called a recursion formula, for calculating any later
term from terms that precede it.


Lecture-1

Using a Recursion Formula
Example Find the first four terms of the sequence a1
= 4; for n>1, an = 2an-1 + 1
Solution We know a1 = 4.
Since an = 2an-1 + 1

a2
a3

2 a2 1 2 9 1 19

a4

2 a1 1 2 4 1 9
2 a3 1 2 19 1 39
Lecture-1

Applications of Sequences
Example The winter moth population in
thousands
per acre in year n, is modeled by

a1 1, an

2.85an

1

2
.19an 1

for n > 2

(a) Give a table of values for n = 1, 2, 3, …, 10

(b) Graph the sequence.


Lecture-1

Applications of Sequences
Solution
(a)
n
an
n
an
(b)

1
1
7
9.31

2
2.66
8
10.1

3
6.24
9
9.43

4
5
6
10.4 9.11 10.2
10
9.98
Note the population
stabilizes near a value
of 9.7 thousand insects
per acre.


Lecture-1

Assignment-1
 Let an and bn be sequences of real numbers and let A and

B be real numbers. The following rules hold if
and lim tn bn B
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Sum Rule: lim tn (an bn ) A B
Difference Rule: lim tn (an bn ) A B
lim tn (an .bn ) A.B
Product Rule:
Constant Multiple Rule: lim tn (k .bn ) k .B
Quotient Rule: lim tn an A if B 0
bn


lim tn

an

A

(any number of k)

B

Lecture-1

Infinite sequence & series 1st lecture

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