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Alternating Series and Absolute Convergence Alternating Series Test: The alternating series (1) n 1 n 1 a n a1 a 2 a3 a 4 (1) n 1 a n is convergent if the following two conditions are satisfied: (i) a k a k 1 0 for every k (ii) lim n an 0 Example 1: Determine whether the alternating series converges or diverges. (a) (1) n1 n Definition: A series | a n 2n 4n 2 3 a (b) (1) n n n 1 2n 4n 3 is absolutely convergent if the series || a1 | | a2 | | an | is convergent. Example 2: Show that the following alternating series is absolutely convergent. 1 1 1 1 1 1 (1) n 1 n 2 3 4 n Show that this series is (a) convergent (b) not absolutely convergent Example 3: The alternating harmonic series is Definition: A series divergent. a Theorem: If the series n (1) is conditionally convergent if a n 1 n a is absolutely convergent, then 1 1 1 2 3 4 a be the series 2 2 2 2 Determine whether a converges or diverges. Example 4: Let n 1 n a is convergent and n is convergent. 1 1 1 1 6 7 8 5 2 2 2 2 n Example 5: Determine whether the following series is convergent or divergent: sin 2 sin 3 sin n sin 1 2 2 2 2 3 n | a n | is A series may be classified in exactly one of the following ways: (1) absolutely convergent (ii) conditionally convergent Ratio Test For Absolute Convergence: Let lim n (i) (ii) (iii) a n (iii) divergent be a series of nonzero terms, and suppose a n 1 L an If L<1, the series is absolutely convergent a If L>1, or lim n n 1 , the series is divergent. an If L = 1, apply a different test; the series may be absolutely convergent, conditionally convergent, or divergent. Example 6: Determine whether the following series is absolutely convergent, conditionally 2 n n 4 convergent, or divergent: (1) 2n n 1 Power Series and Interval of Convergence: Example 1: Find all values of x for which the following power series is absolutely convergent (Interval of convergence): 1 2 n 1 x 2 x2 n xn 5 5 5 Example 2: Find all values of x for which the following power series is absolutely convergent: 1 1 1 1 x x2 xn 1! 2! n! Example 3: Find all values of x for which n! x n is convergent: Theorem: (i) If a power series an x n converges for a nonzero number c, then it is absolutely convergent whenever | x || c | . (ii) If a power series an x n diverges for a nonzero number d, then it diverges whenever | x || d | . Theorem: (i) (ii) (iii) If an x n is a power series, then exactly one of the following is true: The series converges only if x 0. The series is absolutely convergent for every x. There is a number r>0 such that the series is absolutely convergent if x is in the open interval (-r,r) and divergent if x<-r or x>r. The number r is the radius of convergence of the series. Either convergence or divergence may occur at –r or r, depending on the nature of the series. The totality of numbers for which a power series converges is called its interval of convergence. If the radius of convergence r is positive, then the interval of convergence is one of the following (-r,r), (-r,r], [-r,r), [r,r] To determine which of these intervals occurs, we must conduct separate investigations for the numbers x r and x r . The radius of convergence can be denoted 0 or (Ex 2 and 3 respectively). (note: save this for next class) Example 4: Find the interval of convergence and the radius of 1 n x . convergence of the power series n n 1 k 1 n x for k = 3, 4, 5, and 6. Use a graphing utility to graph the polynomials Pk ( x) n n 1 Example 5: Find the interval of convergence and radius of convergence of the series 1 1 1 1 ( x 3) ( x 3) 2 (1) n ( x 3) n 2 3 n 1