Calculus A power Series in x - a is an Infinite Series

Definition: A power series in x − a is an infinite series of the form P∞ n 2 n=0 cn (x − a) = c0 + c1 (x − a) + c2 (x − a) + · · · . In calculus, you learned many tests to determine where an infinite series converges or diverges. Two of those tests are the Ratio Test and the Alternating Series Test. Ratio Test: Let P an be a series of nonzero terms, and suppose limn→∞ | aan+1 | = L. n (i) If L < 1, the series is absolutely convergent. | = ∞, the series is divergent. (ii) If L > 1 or limn→∞ | aan+1 n (iii) If L = 1, apply a different test; the series may be absolutely convergent, conditionally convergent, or divergent. Definition: An alternating series is a series whose terms are alternately positive and negative. Alternating Series Test: If the alternating series P∞ n−1 bn = b1 − b2 + b3 − b4 + b5 − b6 + ...... bn > 0 n=1 (−1) satisfies (i) bn+1 ≤ bn for all n and (ii) limn→∞ bn = 0 then the series is convergent. Definition: The interval of convergence for a power series is the set of all real numbers x for which the series converges. Definition: The radius R of the interval of convergence of a power series is called its radius of convergence. Exercise 1. Determine the convergence set of P∞ (−2)n n=0 n+1 (x − 3)n (Nagle Section 8.2 Example 1) Exercise radius ofP convergence for the power series. (#2,4,6) P∞ 2. Find the interval P∞ and n ∞ (a) n=1 n12 xn (b) n=0 5n! xn (c) k=0 k!(x − 1)k Class Exercise 1. Find the interval and radius of convergence for the given power series. (#8,10) n P∞ P∞ (b) n=0 (−1) x2n+1 (a) k=0 3−k (4x − 5)k 9n Maclaurin series for f (x): If a function f has a power series representation P∞ f (x) = n=0 an xn with radius of convergence r > 0, then f (k) (0) exists for every positive integer k and an = f (n) (0)/n!. Thus, f (x) = f (0) + f ′ (0)x + f ′′ (0) 2 2 x + ··· + Exercise 3. Find the Maclaurin series for f (x) = ex . Common Maclaurin Series ex = P∞ cos x = 1 n=0 n! P∞ n=0 xn (−1)n (2n)! x2n f (n) (0) n n! x + ···. sin x = (−1)n n=0 (2n+1)! P∞ (−1)n n=0 2n+1 tan−1 x = P∞ cosh x = P∞ sinh x = P∞ 1 n=0 (2n)! P∞ P∞ xn = n=0 n=1 x2n+1 x2n 1 n=0 (2n+1)! ln(1 + x) = 1 1−x x2n+1 x2n+1 (−1)n+1 n xn Exercise 4. Use an appropriate series in the table to find the Maclaurin series for the function xe3x . (#12) Class Exercise 2. Use an appropriate series in the table to find the Maclaurin series of the given x function: (a) 1+x (b) sin x2 . (#14, 16) 2 Definition: A function f is said to be analytic at a point a if it can be represented by a power series in x − a with either a positive or an infinite radius of convergence. Exercise 5. f (x) = e−x cos x is analytic at a = 0. Use an appropriate series in the table and multiplication to find the first four nonzero terms of the Maclaurin series of the function. (#20) Class Exercise 3. f (x) = tan x is analytic at a = 0. Use an appropriate series in the table and long division to find the first four nonzero terms of the Maclaurin series of the function. (#22) Exercise 6. Rewrite the expression P∞ n=1 ncn xn−1 + 3 P∞ n=0 cn xn+2 using a single power series whose general term involves xk . (#26) Class Exercise 4. Rewrite the expression P∞ P∞ n−2 + n=0 cn xn+2 n=2 n(n − 1)cn x using a single power series whose general term involves xk . (#28) Exercise 7. Verify by direct substitution that the power series y = of the differential equation P∞ n=0 (−1)n x2n is a solution (1 + x2 )y ′ + 2xy = 0. (#32) Class Exercise 5. Verify by direct substitution that the power series y = solution of the differential equation (−1)n 2n n=0 22n (n!)2 x P∞ xy ′′ + y ′ + xy = 0. (#34) Exercise 8. Find a power series solution P∞ n=0 cn xn of the differential equation 4y ′ + y = 0. (#36) Class Exercise 6. Find a power series solution P∞ n=0 cn xn of the differential equation (1 + x)y ′ + y = 0. (#38) is a