Chapter 4 Sequences and Series

Transcription

1 Chapter 4 Sequences and Series 4.1 Sequence Review Sequence: a set of elements (numbers or letters or a combination of both). The elements of the set all follow the same rule (logical progression). The number of elements in the set can be either finite or infinite. A sequence is usually represented by using brackets of the form { } and placing either the rule or a number of the elements inside the brackets. Some simple examples of sequences are listed below. The alphabet: The set of natural numbers less than or equal to 50: The set of all natural numbers: The set {a n } where NOTE: As this last example suggests, the general element of the sequence is normally represented by using a subscripted letter, with the range of the subscript being the natural (counting) numbers unless otherwise noted. As the value of the subscript increases (tends to infinity) the general term of the sequence may or may not have a single value. If it does have a single finite value, then the sequence is said to converge, and the sequence converges to that single finite value. If the limiting value is infinite or if no limiting value exists then the sequence is said to diverge. 4.2 Series Series: A series is a sum of elements. The sum can be finite or it can be infinite. The elements of the series can be either numbers or letters or a combination of both. A series can be represented (1) by listing a number of elements along with the appropriate sign (+ or ) between the elements OR (2) by using what is called sigma notation with only the general term and the range of summation indicated. Examples: (1) (2) Both of these examples represent the same series. 52

2 Convergent Series: A series of which the sum exists. Divergent Series: A series of which the sum does not exist. Theorem: A series converges iff the associated sequence of partial sums represented by {S k } converges. The element S k in the sequence above is defined as the sum of the first k terms of the series. Common series includes the following categories: (1) (2) (3) Power Series Power Series: Any series of the form n a ( ) n 0 n x c is called a power series. n n f ( x) a n n ( x c) 0 Where a n represents the coefficient of the nth term, c is a constant, and x varies around c (for this reason one sometimes speaks of the series as being centered at c). It is a series in powers of (x c), where c is called the centre of the series. One of the most important examples of a power series is geometric series formula which is valid for x < 1, as are the exponential function formula and the sine formula valid for all real x. You will find that these power series are also examples of Taylor series. 53

3 4.2.2 Taylor Series Taylor Series: A series used to represent a function through summation of terms (infinite) based on the function s different orders of derivatives at a single point. Consider a function f (x) for which the first n-th order derivatives are continuous on the closed interval [x o, x], then ( n) n f ( xo ) n f ( x) ( x x n 0 o ) = n! is the n-th degree Taylor Series polynomial of f (x) centered at x= x o. In above, the factorials: If in the Taylor Series the value of x o is set to zero, then it is called a Maclaurin Series. Example 4.1 Find Taylor series for (1) f (x) = 3; (2) f (x) = x +3; (3) f (x) = 2 x 2 + x +3 with center at x= 0. 54

4 Example 4.2 Find the Taylor series expansion of the function y= e x by 1 st 8 terms with center at x= 0. 55

5 Example 4.3 Find the Taylor series of the function f (x) = sin(x) centered at x= Differentiation and Integration We can also find the Taylor series of a function based on the series of the function s derivative or integral. 56

6 Example 4.4 Find the Maclaurin series of the function f (x) = cos(x). From example 4.3, we know the Maclaurin series of sin(x) is Example 4.5 Find the Maclaurin series of the function f (x) = 1/(1-x) 3. 57

7 Example 4.6 Find the Taylor series for f (x) = ln x centered at x= 1. 58

8 4.2.4 Fourier Series Fourier Series: A series defining a periodic function through summation of simple functions (generally orthogonal trigonometric function such as sine and cosine functions). Fourier series are widely used in signal processing, solving differential equations and so on. (1) Suppose f(x) is periodic with period 2π, it can be represented in Fourier series as f(x)= where 59

9 a 0 = a n = b n = (2) If f(x) is periodic with period 2P, then it can be represented in Fourier series as f(x)= where a 0 = a n = b n = Example 4.7 Find the Fourier series of the sawtooth waveform shown below. 60

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