POWER SERIES; INTERVAL OF CONVERGENCE
10. POWER SERIES; INTERVAL OF CONVERGENCE
We have been discussing series whose terms were constants. Even more important and useful are series whose terms are functions of x. There are many such series, but in this chapter we shall consider series in which the nth term is a constant times x n
or a constant times (x −a) n where a is a constant. These are called power series, because the terms are multiples of powers of x or of (x − a). In later chapters we
shall consider Fourier series whose terms involve sines and cosines, and other series (Legendre, Bessel, etc.) in which the terms may be polynomials or other functions.
By definition, a power series is of the form
or (10.1) n=0
a x n =a +a x+a x 2 +a x n 3 0 1 2 3 +···
n (x − a) =a 0 +a 1 (x − a) + a 2 (x − a) +a 3 (x − a) +···,
n=0
where the coefficients a n are constants. Here are some examples:
x 2 x 3 x 4 (−1) n+1 x n
(10.2b) x−
2 3 4 n x 3 x 5 x 7 (−1) n+1 x 2n−1
(10.2c) x−
Whether a power series converges or not depends on the value of x we are considering. We often use the ratio test to find the values of x for which a series converges. We illustrate this by testing each of the four series (10.2). Recall that in the ratio test we divide term n + 1 by term n and take the absolute value of this ratio to get ρ n , and then take the limit of ρ n as n → ∞ to get ρ.
Example 1. For (10.2a), we have
(−x) n+1
(−x) n
2 n+1 ÷ 2 n
x ρ=
Section 10 Power Series; Interval of Convergence
The series converges for ρ < 1, that is, for |x/2| < 1 or |x| < 2, and it diverges for |x| > 2 (see Problem 6.30). Graphically we consider the interval on the x axis between x = −2 and x = 2; for any x in this interval the series (10.2a) converges. The endpoints of the interval, x = 2 and x = −2, must be considered separately. When x = 2, (10.2a) is
which is divergent; when x = −2, (10.2a) is 1 + 1 + 1 + 1 + · · · , which is divergent. Then the interval of convergence of (10.2a) is stated as −2 < x < 2.
Example 2. For (10.2b) we find
The series converges for |x| < 1. Again we must consider the endpoints of the interval of convergence, x = 1 and x = −1. For x = 1, the series (10.2b) is
1− 1 2 + 1 3 − 1 4 + · · · ; this is the alternating harmonic series and is convergent. For x = −1, (10.2b) is −1 − 1 2 − 1 3 − 1 4 − · · · ; this is the harmonic series (times −1) and is divergent. Then we state the interval of convergence of (10.2b) as −1 < x ≤ 1. Notice carefully how this differs from our result for (10.2a). Series (10.2a) did not converge at either endpoint and we used only < signs in stating its interval of convergence. Series (10.2b) converges at x = 1, so we use the sign ≤ to include x = 1. You must always test a series at its endpoints and include the results in your statement of the interval of convergence. A series may converge at neither, either one, or both of the endpoints.
Example 3. In (10.2c), the absolute value of the nth term is |x 2n−1 /(2n − 1)!|. To get term n + 1 we replace n by n + 1; then 2n − 1 is replaced by 2(n + 1) − 1 = 2n + 1, and the absolute value of term n + 1 is
x 2n+1 (2n + 1)!
Thus we get
x 2n+1
x 2n−1
(2n + 1)! ÷ (2n − 1)!
(2n + 1)(2n)
x 2 ρ = lim n→∞ (2n + 1)(2n)
Since ρ < 1 for all values of x, this series converges for all x. Example 4. In (10.2d), we find
(x + 2) n+1
(x + 2) n
n+2
n+1
√ n+1
ρ = lim n→∞
n+2
22 Infinite Series, Power Series Chapter 1
The series converges for |x+2| < 1; that is, for −1 < x+2 < 1, or −3 < x < −1. If x = −3, (10.2d) is
2 3 4 which is convergent by the alternating series test. For x = −1, the series is
2 3 n=0 n+1
which is divergent by the integral test. Thus, the series converges for −3 ≤ x < 1.
PROBLEMS, SECTION 10
Find the interval of convergence of each of the following power series; be sure to investigate the endpoints of the interval in each case.
X ∞ n n 1. (−1) x
X ∞ n n X (2x) n
X ∞ n x n 4. n 2 5. 2 (−1) 6. n=1 2 n
n=1 (n!)
(2n)!
n=1
X ∞ x 3n X n x n X 7. 8. (−1) √
X ∞ (−1) x 2n 1 “
n(−2x)
13. n(−x)
17. (−1) n (x + 1) n
X (−2) n (2x + 1) n
The following series are not power series, but you can transform each one into a power series by a change of variable and so find out where it converges.
19. ∞ P 8 − n (x 2 − 1) n Method: Let y = x 2 P − 1. The power series ∞ 8 − n 0 y 0 n converges for |y| < 8, so the original series converges for |x 2 − 1| < 8, which means |x| < 3.
X ∞ n 2 n 2 2n X n x n/2 20. (−1)
X ∞ n!(−1) n X 3 n (n + 1)
3 25. (sin x) n
0 3 +n
Section 12 Expanding Functions in Power Series
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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