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9.4 Radius of Convergence Quick Review Find the limit of the expression as n . Assume x remains fixed as n changes. nx n 1 n x 3 2. n n 1 1. 2 x n! n+1 x 4. 2n 3. 4 2 4 2 2x 1 5. 2 2x 1 n n 1 n 1 n Quick Review Let a be the nth term of the first series and b the nth term of the n n second series. Find the smallest positive integer N for which a b for all n N . Identify a and b . n n n 6. 5n, n 7. n , 5 5 2 n 8. ln n, n 1 9. , n 10 -3 n n Quick Review Solutions Find the limit of the expression as n . Assume x remains fixed as n changes. nx |x| n 1 n x 3 2. | x 3| n n 1 1. 2 x 0 n! n+1 x 4. 2n 3. 4 2 2 x /16 4 2 2x 1 5. 2 2x 1 n n 1 n 1 n | 2x 1 | / 2 Quick Review Solutions Let a be the nth term of the first series and b the nth term of the n n second series. Find the smallest positive integer N for which a b for all n N . Identify a and b . n n n n 6. 5n, n a n , b 5n, N 6 7. n , 5 a 5 , b n , N 6 2 2 n 5 n n n n 8. ln n, 1 9. , 10 1 10. , n n a n 1 n! n 2 n -3 5 n n , b ln n, N 1 n 1 1 , b , N 25 10 n! 1 a , b n , N 2 n a n n n 3 n 2 n What you’ll learn about Convergence nth-Term Test Comparing Nonnegative Series Ratio Test Endpoint Convergence … and why It is important to develop a strategy for finding the interval of convergence of a power series and to obtain some tests that can be used to determine convergence of a series. The Convergence Theorem for Power Series There are three possibilities for c x a with respect to convergence: n 0 n n 1. There is a positive number R such that the series diverges for x a R but converges for x a R. The series may or may not converge at either of the endpoints x a R amd x a R. 2. The series converges for every x ( R ). 3. The series converges at x a and diverges elsewhere ( R 0). The nth-Term Test for Divergence a diverges if lim a fails to exist or is different from zero. n 1 n n n The Direct Comparison Test Let a be a series with no negative terms. n (a) a converges if there is a convergent series c with a c for all n N , for some integer N . n n n n (b) a diverges if there is a divergent series d with a d for all n N , for some integer N . n n n n Example Proving Convergence by Comparison x Prove that converges for all real x. 2n n 0 n ! 2 2n x Let x be any real number. The series has no negative terms. n ! n 0 x x x x . Recognize as the Taylor series n! n ! n ! n ! 2n For any n, 2 2 2n n 2 n 2 n 0 x2 x2 for e , which we know converges to e for all real numbers. Since 2n x the e series dominates term by term, the latter series must also n ! x2 n 0 2 converge for all real numbers by the Direct Comparison Test. Absolute Convergence If the series a of absolute values converges, then a n converges absolutely. n Absolute Convergence Implies Convergence If a converges, then a converges. n n Example Using Absolute Convergence sin x Show that converges for all x. n n! n 0 n sin x Let x be any real number. The series has no negative terms, n! 1 and it is term-by-term less than or equal to the series , which converges n! sin x to e. Therefore, converges by direct comparison. n! sin x Since converges absolutely, it converges. n! n 0 n 0 n n 0 n n 0 The Ratio Test Let a be a series with positive terms, and with lim n n a L. a n 1 n Then, (a) the series converges if L 1, (b) the series diverges if L 1, (c) the test is inconclusive if L 1. Example Finding the Radius of Convergence nx Find the radius of convergence of . n n 0 10 n Check for absolute convergence using the Ratio Test. a n 1 x lim lim a 10 n 1 n n n n 1 n 1 n 10 nx n x n 1 x lim n 10 10 x Setting 1, we see that the series converges absolutely for 10 x 10. 10 The series diverges for x 10 and x 10. n The radius of convergence is 10. Example Determining Convergence of a Series 2 Determine the convergence or divergence of the series . n n 0 Use the Ratio Test: 2 n 1 a 2 3 1 lim lim 3 1 lim 2 a 3 1 2 3 1 3 1 lim 2 3 1 1 1 lim 2 3 1 3 3 2 2 The series converges because 1. 3 3 n 1 n 1 n n 1 n n n n n n n n n 1 n n n n 1 n 3 1 n Pg. 386, 7.1 #1-25 odd