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Infinite sequence & series 1st lecture
1.
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
2.
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 . Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
3.
Course Description 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 Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
4.
Syllabus 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 Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
5.
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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
6.
Sequences Copyright © 2013
CHEP, P. U. Lahore. Lecture-1
7.
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 Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
8.
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}. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
9.
Sequences A sequence is
a function that has a set of natural numbers as its domain. f (x) notation is not used for sequences. Write an f (n) Sequences are written as ordered lists a1 , a2 , a3 , ... a1 is the first element, a2 the second element, and so on Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
10.
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} Copyright © 2013 CHEP, P. U. Lahore. are Lecture-1
11.
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,. . . . . Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
12.
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) Copyright © 2013 CHEP, P. U. Lahore. 2 / 3, a2 4 / 5, a4 (2 1) /(2 2) 3/ 4 (4 1) /(4 2) 5 / 6 Lecture-1
13.
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 Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
14.
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, … Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
15.
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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
16.
Convergent and Divergent
Sequences 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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
17.
Convergent and Divergent
Sequences 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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
18.
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 The value(s) of the initial term(s) The rule, called a recursion formula, for calculating any later term from terms that precede it. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
19.
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 Copyright © 2013 CHEP, P. U. Lahore. 2 a1 1 2 4 1 9 2 a3 1 2 19 1 39 Lecture-1
20.
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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
21.
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. Copyright © 2013 CHEP, P. U. Lahore. Lecture-1
22.
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 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 Copyright © 2013 CHEP, P. U. Lahore. lim tn an A (any number of k) B Lecture-1
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