Fourier Analysis. Lecture 1: Sequences and Series. v1 by G. M. Chávez-Campos. Updated: 2016/06/27. Instituto Tecnológico de Morelia

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1 Fourier Analysis Lecture 1: Sequences and Series v1 by G. M. Chávez-Campos Updated: 2016/06/27 Instituto Tecnológico de Morelia

2 Course information

3 Course information La asignatura le aporta al estudiante herramientas anaĺıticas, relacionadas con matemáticas avanzadas, que le permiten resolver problemas de ecuaciones y circuitos relacionados con la electrónica analógica y digital. Didactics El estudiante debe realizar actividades que le permitan desarrollar las competencias necesarias para comprender el uso de series, sucesiones y transformadas de Fourier utilizadas en la solución de problemas en electrónica analógica y digital.

4 Course Target El alumno debe comprender el uso de series, sucesiones y transformadas de Fourier utilizadas en la solución de problemas en electrónica analógica-digital.

5 Course topics Lecture 1: Sequences, series and Fourier series. Lecture 2: Fourier Transform. Lecture 3: Inverse continuos Fourier transform. Lecture 4: Discrete Fourier transform. Lecture 5: Inverse discrete Fourier transform. Lecture 6: Fast Fourier Transform.

6 Overview 1. Course information 2. Sequences 2.1 Introduction: practical cases 2.2 Type of sequences 3. Series 3.1 Finite and infinite series 3.2 Convergency 3.3 The kth term theorem 3.4 Special series Arithmetic series Harmonic series Geometric series

7 Sequences

8 Practical situations Try to solve the next practical cases: Case 1: Consider that you are throwing an object from the first floor of a building, the object falls 16ft vertical distance; from second floor, the object falls 48ft; third floor, 80ft; and so on. How many distance will falls the object in floor 4, 5 and 6? Case 2: A bacterium is able to reproduce itself by double every 15 minutes. How many bacteria will be after 2 hours?

9 Sequences Definition A sequence is a number of people or things of a similar kind following one after the other. Also the sequence could be a group of real numbers. This sequence is said to be finite because there is a first and last number. If the set of numbers which defines a sequence does no have both a first and last number, the sequence is said to be infinite.

10 Sequences cont. Finite 1,2,4,8-24,-20,-16,...,0

11 Sequences cont. Finite 1,2,4,8-24,-20,-16,...,0 Infinite 1,3,6,... 0,5,10,15,20,...

12 Recursive Definition A recursive succession commonly offers the next information: First term or first terms, it is specified some way to obtain next terms, using the first terms

13 Hands on... The student may calculate the first five terms for the succession a 1 = 5 and a n = a n for n {2, 3, 4...}: a 1 = 5 a 2 a 3 a 4 a 5

14 Exercise 2 The student may calculate the first five terms for the succession a 1 = 4 and a n = 1 2 a n 1 for n 2: a 1 = 4 a 2 a 3 a 4 a 5

15 Series

16 Series Definition If {U n } is a sequence and S n = n U i = U 1 + U 2 + U U n i=1 then the sequence {S n } is called an infinite series. The numbers U 1, U 2, U 3,... are called the terms.

17 Series: Finite and infinite series S n = U n = U 1 + U 2 + U U n + (1) n=1 S k = k U i = U 1 + U 2 + U U k (2) i=1 where (1) is an infinity series and (2) is kth partial term of a given series. sy

18 Exercise 3: Given the infinity series: n=1 1 n(n + 1) find the first four elements of the sequence of partial sums {s n }, and find a formula for s n in terms of n.

19 Convergent series Definition Let n=1 U n be a given infinite series, and let {S n } be the sequence of partial sums defining this infinity series. Then if lim S n exists and is equal to S, we say that the given series is convergent and that S is the sum of the given infinity series. If lim S n does no exist, the series is said to be divergent and the series does not have a sum.

20 Exercise 4: Given the infinity series: + n=1 1 n(n + 1)

21 Exercise 4: Given the infinity series: + n=1 lim S n = n + 1 n(n + 1) lim n + n n + 1

22 Exercise 4: Given the infinity series: + n=1 lim S n = n + 1 n(n + 1) n=1 lim n + n n + 1 n n + 1

23 The k th term theorem Theorem If the infinity series + n=1 U n is convergent If the infinity series + n=1 U n is divergent lim U n = 0 (3) n + lim U n 0 (4) n +

24 Exercise 5: + n=1 + n=1 n n 2 (5) 3 ( 1) n+1 (6)

25 Harmonic series Note that the Theorem (3) not always is true. In other words, it is possible to have a divergent series for which lim n + U n = 0. An example of such a series is the one known as the harmonic, which is + n=1 1 n = n (7)

26 Does the harmonic series converges? Theorem Let {S n } be the sequence of partial sums for a given convergent series + n=1 U n. Then for any ɛ > 0 there exist a number N such that S R S T < ɛ, whenever R > N and T > N

27 Does the harmonic series converges? Theorem Let {S n } be the sequence of partial sums for a given convergent series + n=1 U n. Then for any ɛ > 0 there exist a number N such that S R S T < ɛ, whenever R > N and T > N Copyright by [2]

28 Geometric series The geometric series is a sires of the form + n=1 Lets try to understand... ar n 1 = a + ar + ar ar n 1 + (8)

29 Geometric series The geometric series is a sires of the form + n=1 Lets try to understand... ar n 1 = a + ar + ar ar n 1 + (8) S n = a(1 r n ) 1 r (9)

30 Geometric series convergency Theorem The geometric series converges to the sum a/(1 r) if r < 1, and the geometric series diverges if r >= 1

31 Exercise 6: Express the decimal number as a common fraction.

32 Two series theorem Theorem If + n=1 a n and + n=1 b ntwo infinite series, differing only in their first m terms, then either both series converge or both series diverge.

33 Two series theorem Theorem If + n=1 a n and + n=1 b ntwo infinite series, differing only in their first m terms, then either both series converge or both series diverge. Theorem If the series + n=1 a n is convergent and its sum is S, then the series + n=1 ca n is also convergent and its sum is c S If the series + n=1 a n is divergent, then the series + n=1 ca n is also divergent

34 Exercise 7: Determine whether the infinity series is convergent or divergent + n=1 1 4n

35 The integral test The theorem known as the integral test makes use of the theory of improper integrals to test an infinite series of positive terms for convergence.

36 The integral test The theorem known as the integral test makes use of the theory of improper integrals to test an infinite series of positive terms for convergence. Theorem Let f be a function which is continuous, decreasing, and positive valued for all x >= 1, then the infinite series + n=1 is convergent if the improper integral f (n) = f (1) + f (2) + + f (n) f (x)dx exists, and is divergent if the improper integral increase without bound.

37 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus:

38 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent.

39 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent.

40 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent. If L = 1, the series may be absolutely convergent, conditionally convergent, or divergent.

41 Theorem Let a n be an infinite series of nonzero terms and let L to be calculated by (10) lim a n+1 n + a n = L (10) thus: If L < 1, the series is absolutely convergent. If L > 1, or L, the series is divergent. If L = 1, the series may be absolutely convergent, conditionally convergent, or divergent.

42 Exercise 8: Determine whether the following series are absolutely convergent, conditionally convergent, or divergent. ( 1) n 3n n! (11) 3 n n 2 (12) n=1 n=1

43 Power series

44 Power series Theorem Let x be a variable. A power series in x is a series of the form [2] a n x n = a 0 + a 1 x + a 2 x a n x n + (13) n=0 where each a n is a real number.

45 Exercise 9: Find all values of x for which the following power series is absolutely convergent x x n 5 n x n + (14) ! x + 1 2! x n! x n + (15)

46 Power series convergency theorem Theorem If a n x n is a power series, then precisely one of the following is true The series converges only if x = 0, The series is absolutely convergent for all x, There is a positive number r such that the series is absolutely convergent if x < r and divergent if x > r

47 convergency radius

48 Power series x c Theorem Let c be a real number and x a variable. A power series in x c is a series of the form + n=0 a n (x c) n = a 0 + a 1 (x c) + a 2 (x c) a n (x c) n + where each a n is a real number. (16)

49 Power series convergency theorem Theorem If a n (x c) n is a power series, then precisely one of the following is true The series converges only if x c = 0, The series is absolutely convergent for all x, There is a positive number r such that the series is absolutely convergent if x c < r and divergent if x c > r

50 convergency radius

51 Exercise 10: Find the interval of convergence of (x 3) (x 3)2 + (17)

52 Power series representations of function A power series a n x n may be used to define a function of f whose domain is the interval of convergence of the series. Specifically, for each x in this interval we let f (x) equal the sum of the series, that is f (x) = a 0 + a 1 x + a 2 x a n x n + (18) If a function f is defined in this way we say that a n x n is a power series representative for f (x). This allows us to find values in a new way. Specifically, if c is the interval of convergence, thus f (c) can be found or approximated be the series...

53 Exercise 11: Find a function f that is represented by the power series ( 1) n x n (19) n=0

54 Theorem Suppose a power series a n x n has a nonzero radius of convergence r and let the function f be defined by f (x) = a n x n (20) n=0 for every x in the interval of convergence. If r < x < r, then: f (x) = D x (a n x n ) = n=0 na n x (n 1) (21) n=1 = a 1 + 2a 2 x + 3a 3 x na n x (n 1) +

55 Integral Theorem x 0 f (t)dt = n=0 x 0 (a n t n )dt = n=0 a n n + 1 x n+1 (22) = a 0 x a 1x a 2x n + 1 a nx n+1 +

56 Tylor and Maclaurin Series

57 Tylor and Maclaurin Series Suppose a function f is represented by a power series in x c, such that f (x) = + n=0 a n (x c) n = a 0 +a 1 (x c)+a 2 (x c) 2 +a 3 (x c) 3 + where the domain of f is an open interval containing c (23)

58 f (x) = na n (x c) n 1 (24) n=1 = a 1 + 2a 2 (x c) + 3a 3 (x c) 2 + 4a 4 (x c) 3 + f (x) = n(n 1)a n (x c) n 2 (25) n=2 = 2a 2 + (3 2)a 3 (x c) + (4 3)a 4 (x c) 2 + f (x) = n(n 1)(n 2)a n (x c) n 3 (26) n=3 = (3 2)a 3 + (4 3 2)a 4 (x c) +

59 and, for every positive integer k, f (k) (x) = n(n 1) (n k + 1)a n (x c) n k (27) n=k Moreover, each series obtained by differentiation has the same radius of convergence as the original series. Substituting c for x in each of these series representation, we obtain f (c) = a 0 (28) f (c) = a 1 (29) f (c) = 2a 2 (30) f (c) = (3 2)a 3 (31) f (n) (c) = n!a n a n = f (n) (c) n! (32)

60 Taylor series Theorem If f is a function and f (x) = a n (x c) n (33) n=0 for all x in an open interval containing c, then f (x) = f (c) + f (c)(x c) + f (c) 2! (x c) f (n) (c) n! (x c) n (34)

61 Maclaurin Corollary If f is a function and f (x) = a n x n for all x in an open interval ( r, r), then f (x) = f (0) + f (0)x + f (0) 2! x f (n) (0) x n + (35) n!

62 References Louis Leithold The calculus with analytic geometry Harper & Row, Publishers, 1976 Earl W. Swokowski Calculus with analytic geometry Prindle, Weber & Schmidt, 1979 Théorie analytique de la chaleur Jean-Baptiste Joseph Fourier 1822 Sucesiones y Series Roberto O. Rivera Rodríguez 2015