Review of Power Series MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018
Introduction In addition to the techniques we have studied so far, we may use power series to solve ODEs/IVPs. Today we will review the properties of power series and techniques for manipulating power series.
Power Series Definition A power series in x x 0 where x 0 is a constant is an infinite series of the form a n (x x 0 ) n where the a n are constant coefficients. Note: if f (x) = a n (x x 0 ) n then f (x 0 ) = a 0.
Convergence Definition A power series f (x) = x x 0 if exists. a n (x x 0 ) n is said to converge at lim N N a n (x x 0 ) n
Convergence Definition A power series f (x) = x x 0 if exists. a n (x x 0 ) n is said to converge at lim N The power series may converge: only for x = x 0, N a n (x x 0 ) n for all x R, only for x (x 0 ρ, x 0 + ρ) where ρ > 0 (or x (x 0 ρ, x 0 + ρ], or x [x 0 ρ, x 0 + ρ), or x [x 0 ρ, x 0 + ρ]).
Absolute Convergence Definition A power series f (x) = a n (x x 0 ) n is said to converge absolutely at x x 0 if converges. a n (x x 0 ) n = a n x x 0 n Note: absolute convergence implies convergence, but the converse is not true.
Ratio Test Theorem (Ratio Test) Given the power series f (x) = that n a n (x x 0 ) n and supposing lim a n+1 (x x 0 ) a n = L x x 0 1. if L x x 0 < 1 then f (x) = absolutely for x x 0 < 1/L, then a n (x x 0 ) n converges 2. if L x x 0 > 1 then the series diverges, 3. if L x x 0 = 1 the test is inconclusive. Note: ρ = 1/L is the radius of convergence of the power series.
Examples Find the radius of convergence of each of the following power series. x n n!x n x n n!
Examples Find the radius of convergence of each of the following power series. x n (ρ = 1) n!x n x n n!
Examples Find the radius of convergence of each of the following power series. x n (ρ = 1) n!x n (ρ = 0) x n n!
Examples Find the radius of convergence of each of the following power series. x n (ρ = 1) n!x n (ρ = 0) x n n! (ρ = )
Equating Power Series If a n (x x 0 ) n = b n (x x 0 ) n for all x (x 0 ρ, x 0 + ρ) with ρ > 0 then a n = b n for n = 0, 1, 2,....
Arithmetic of Power Series Suppose a n (x x 0 ) n and b n (x x 0 ) n converge to f (x) and g(x) respectively for x x 0 < ρ with ρ > 0. f (x) ± g(x) = (a n ± b n )(x x 0 ) n converges at least for x x 0 < ρ. f (x)g(x) = c n (x x 0 ) n where c n = a 0 b n + a 1 b n 1 + + a n 1 b 1 + a n b 0 and the series converges at least for x x 0 < ρ. f (x) g(x) = d n (x x 0 ) n provided g(x 0 ) 0. The radius of convergence of the series may be less than ρ.
Differentiating Power Series If f (x) = a n (x x 0 ) n has radius of convergence ρ > 0 then f is continuous for x x 0 < ρ, f has derivatives of all orders for x x 0 < ρ. f (x) = f (x) = na n (x x 0 ) n 1 n=1 n(n 1)a n (x x 0 ) n 2 n=2. The derivatives converge absolutely for x x 0 < ρ.
Integrating Power Series If f (x) = a n (x x 0 ) n has radius of convergence ρ > 0 then the power series has radius of convergence ρ > 0. a x n n + 1 (x x 0) n+1 = f (t) dt x 0
Taylor Series If f (x) = and a n (x x 0 ) n has radius of convergence ρ > 0 then f (x) = a n = f (n) (x 0 ) n! f (n) (x 0 ) (x x 0 ) n. n! This is called the Taylor Series for f (x) about x = x 0. The function f is said to be analytic at x = x 0.
Taylor Polynomials Definition The N th partial sum of a Taylor series is polynomial of degree N P N (x) = N f (n) (x 0 ) (x x 0 ) n n! = f (x 0 ) + f (x 0 )(x x 0 ) + + f (N) (x 0 ) (x x 0 ) N N! called the Taylor polynomial of degree N for f expanded about x = x 0.
Taylor s Theorem Theorem (Taylor s Theorem) Suppose that f has N + 1 derivatives on the interval (x 0 r, x 0 + r) for some r > 0. Then for x (x 0 r, x 0 + r), f (x) P N (x) and the error in using P N (x) to approximate f (x) is R N (x) = f (x) P N (x) = f (N+1) (z) (N + 1)! (x x 0) N+1 for some z between x and x 0.
Taylor s Theorem Theorem (Taylor s Theorem) Suppose that f has N + 1 derivatives on the interval (x 0 r, x 0 + r) for some r > 0. Then for x (x 0 r, x 0 + r), f (x) P N (x) and the error in using P N (x) to approximate f (x) is R N (x) = f (x) P N (x) = f (N+1) (z) (N + 1)! (x x 0) N+1 for some z between x and x 0. Remark: R N (x) is called the Taylor remainder.
Taylor s Theorem and Taylor Series Theorem Suppose that f has derivatives of all orders on the interval (x 0 r, x 0 + r) for some r > 0 and that for all x in (x 0 r, x 0 + r). Then for all x in (x 0 r, x 0 + r). lim R N(x) = 0 N f (n) (x 0 ) (x x 0 ) n = f (x) n!
Example Find the Taylor series for f (x) = x about x 0 = 1.
Example Find the Taylor series for f (x) = x about x 0 = 1. f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 0 f (x 0 ) = f (1) = 0.
Example Find the Taylor series for f (x) = x about x 0 = 1. f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 0 f (x 0 ) = f (1) = 0. The Taylor remainder R N (x) = 0 for N > 1,
Example Find the Taylor series for f (x) = x about x 0 = 1. f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 1 f (x 0 ) = f (1) = 0 f (x 0 ) = f (1) = 0 The Taylor remainder R N (x) = 0 for N > 1, thus f (x) =. f (n) (x 0 ) (x x 0 ) n = 1 + (x 1). n!
Example Find the Taylor series for f (x) = e x about x 0 = 0.
Example Find the Taylor series for f (x) = e x about x 0 = 0. f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1.
Example Find the Taylor series for f (x) = e x about x 0 = 0. The Taylor remainder R N (x) = N, f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1. e z (N + 1)! x N+1 0 for all x as
Example Find the Taylor series for f (x) = e x about x 0 = 0. The Taylor remainder R N (x) = N, thus e x = f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1 f (x 0 ) = f (0) = 1. e z (N + 1)! x N+1 0 for all x as f (n) (x 0 ) (x x 0 ) n = n! x n n!
Geometric Series Definition A geometric series with ratio r has the form a + a r + a r 2 + = If a 0 the series converges to otherwise. a 1 r a r n. for r < 1 and diverges
Power Series for Common Functions 1 1 x = e x = sin x = cos x = ln x = x n for 1 < x < 1 x n n! for < x < ( 1) n (2n + 1)! x 2n+1 for < x < ( 1) n (2n)! x 2n for < x < ( 1) n+1 (x 1) n for 0 < x 2 n n=1
Re-indexing a Power Series (1 of 2) Re-index the series starts at 0. n=3 n n 2 + 2 x n so that the summation index
Re-indexing a Power Series (1 of 2) Re-index the series starts at 0. n=3 Replace every n with n + 3. n=3 n n 2 + 2 x n = n+3=3 n n 2 + 2 x n so that the summation index n + 3 (n + 3) 2 + 2 x n+3 = n + 3 (n + 3) 2 + 2 x n+3
Re-indexing a Power Series (2 of 2) Re-index the series below so that the power series may be subtracted. 2a n x n+1 b n x n n=1
Re-indexing a Power Series (2 of 2) Re-index the series below so that the power series may be subtracted. 2a n x n+1 b n x n n=1 2a n x n+1 b n x n = n=1 = = 2a n x n+1 b n+1 x n+1 n+1=1 2a n x n+1 b n+1 x n+1 (2a n b n+1 )x n+1
Example Find the Taylor series about x 0 = 0 for (1 x + x 2 )e x.
Solution (1 x + x 2 )e x = (1 x + x 2 ) = = = = = k=0 k=0 k=0 k=0 x k k! k=0 x k k! k=1 x k k! k=1 x k k! k=0 k=0 (k 1) 2 x k k=0 k! x k k! x k+1 k! + k=0 x k+2 k! x k (k 1)! + kx k k! kx k k! + + k=2 x k (k 2)! k(k 1)x k k=2 k! k(k 1)x k k=0 k!
Homework Read Section 5.1 Exercises: 1 27 odd