Worksheet 8 Solutions - MA 2930, March 16, 2011 Worksheet 8...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MA 2930, March 16, 2011 Worksheet 8 Solutions 1. Find the interval of convergence of the following power series: ( i ) ∞ X n =1 (2 x + 1) n ( n + 1) 2 , ( ii ) ∞ X n =0 ( x- 1) n 3 n , ( iii ) ∞ X n =2 n ! x n n n In each case we just need to apply the ratio test. (i) R = lim n →∞ a n +1 a n = lim n →∞ (2 x +1) n +1 ( n +2) 2 (2 x +1) n ( n +1) 2 = (2 x + 1) lim n →∞ ( n + 1) 2 ( n + 2) 2 = (2 x + 1) The series converges whenever | R | = | 2 x +1 | < 1, i.e., when- 1 < 2 x +1 < 1, i.e., when- 1 < x < 0. Note that the radius of convergence is 1/2. (ii) R = lim n →∞ a n +1 a n = lim n →∞ ( x- 1) n +1 3 n +1 ( x- 1) n 3 n = lim n →∞ x- 1 3 = x- 1 3 The series converges whenever | R | = | x- 1 | / 3 < 1, i.e., when- 3 < x- 1 < 3, i.e., when- 2 < x < 4. The radius of convergence is 3. (iii) R = lim n →∞ a n +1 a n = lim n →∞ ( n +1)! x n +1 ( n +1) n +1 n ! x n n n = x lim n →∞ ( n + 1) n n ( n + 1) n +1 = x lim n →∞ 1 (1 + 1 n ) n = x/e The series converges whenever | R | = | x/e | < 1, i.e., when- e < x < e . The radius of convergence is e. 2. In the following series find the coefficient of x n by rewriting the series with x n as the generic term: ( i ) ∞ X n =1 a n x n +2 + ∞ X n =2 a n +1 nx n- 1 , ( ii ) x ∞ X n =1 na n x n- 1 +(1- x 2 ) ∞ X n =2 n ( n- 1) a n x n- 2 The yoga of index shifting is simply this: if you raise n by a certain amount inside the sum, you should lower the starting value of n by the same amount. (i) ∞ X n =1 a n x n +2 + ∞ X n =2 a n +1 nx n- 1 = ∞ X n =3 a n- 2 x n + ∞ X n =1 a n +2 ( n + 1) x n = ∞ X n =3 a n- 2 x n + 2 a 3 x + 3 a 4 x 2 + ∞ X n =3...
View Full Document

This note was uploaded on 06/10/2011 for the course MATH 2930 taught by Professor Terrell,r during the Spring '07 term at Cornell University (Engineering School).

Page1 / 6

Worksheet 8 Solutions - MA 2930, March 16, 2011 Worksheet 8...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online