More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Calculus II Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 April 21, 2020 (Section 9.5 WebWork Mini-Set due 11:59 PM, April 22 nd ) Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: 1 Apply the Ratio Test on � c k ( x − a ) k by calculating c k +1 ( x − a ) k +1 � � � � lim for x � = a . � � c k ( x − a ) k k → + ∞ � � Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: 1 Apply the Ratio Test on � c k ( x − a ) k by calculating c k +1 ( x − a ) k +1 � � � � lim for x � = a . � � c k ( x − a ) k k → + ∞ � � 2 The power series converges absolutely when the above limit is less than 1 and diverges when the above limit is greater than 1. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: 1 Apply the Ratio Test on � c k ( x − a ) k by calculating c k +1 ( x − a ) k +1 � � � � lim for x � = a . � � c k ( x − a ) k k → + ∞ � � 2 The power series converges absolutely when the above limit is less than 1 and diverges when the above limit is greater than 1. 3 The ratio test is inconclusive for any x values where the above limit is exactly 1. In this case, you will need to apply a different convergence test for such x values individually. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series Process for Finding Radii and Intervals of Convergence The general procedure to find the radius and interval of convergence of a power series is as follows: 1 Apply the Ratio Test on � c k ( x − a ) k by calculating c k +1 ( x − a ) k +1 � � � � lim for x � = a . � � c k ( x − a ) k k → + ∞ � � 2 The power series converges absolutely when the above limit is less than 1 and diverges when the above limit is greater than 1. 3 The ratio test is inconclusive for any x values where the above limit is exactly 1. In this case, you will need to apply a different convergence test for such x values individually. Reminder: the interval of convergence is centered about x = a and contains the interval ( a − R , a + R ) at minimum. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: 3 n +1 n =0 ∞ n ( x + 2) n � Find the radius and interval of convergence of . 3 n +1 n =0 Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: 3 n +1 n =0 ∞ n ( x + 2) n � Find the radius and interval of convergence of . 3 n +1 n =0 Apply the Ratio Test. ( n + 1)( x + 2) n +1 3 n +1 � � � ( n + 1)( x + 2) � � � � � lim · � = lim � � � � 3 n +2 n ( x + 2) n 3 n n → + ∞ n → + ∞ � � � = | x + 2 | n + 1 · lim 3 n n → + ∞ = | x + 2 | 3 Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 So the power series converges absolutely if | x + 2 | < 1 ⇒ | x + 2 | < 3 ⇒ − 3 < x + 2 < 3 ⇒ − 5 < x < 1 . 3 (Since the power series is centered about x = − 2 , it follows that the radius of convergence R is 3.) Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 So the power series converges absolutely if | x + 2 | < 1 ⇒ | x + 2 | < 3 ⇒ − 3 < x + 2 < 3 ⇒ − 5 < x < 1 . 3 (Since the power series is centered about x = − 2 , it follows that the radius of convergence R is 3.) The power series diverges if | x + 2 | > 1 , i.e. x < − 5 or x > 1 . 3 Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 So the power series converges absolutely if | x + 2 | < 1 ⇒ | x + 2 | < 3 ⇒ − 3 < x + 2 < 3 ⇒ − 5 < x < 1 . 3 (Since the power series is centered about x = − 2 , it follows that the radius of convergence R is 3.) The power series diverges if | x + 2 | > 1 , i.e. x < − 5 or x > 1 . 3 For | x + 2 | = 1 (that is, when x = − 5 or x = 1), the Ratio 3 Test fails. So we need to investigate these cases separately. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 x = − 5 : The power series becomes ∞ ∞ ∞ n ( − 5 + 2) n n · ( − 3) n n · ( − 1) n � � � = = , 3 n +1 3 n +1 3 n =0 n =0 n =0 which diverges since the terms do not converge to 0. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 x = − 5 : The power series becomes ∞ ∞ ∞ n ( − 5 + 2) n n · ( − 3) n n · ( − 1) n � � � = = , 3 n +1 3 n +1 3 n =0 n =0 n =0 which diverges since the terms do not converge to 0. x = 1 : The power series becomes ∞ ∞ ∞ n (1 + 2) n n · 3 n n � � � = 3 n +1 = 3 , 3 n +1 n =0 n =0 n =0 which diverges since the terms do not converge to 0. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series ∞ n ( x +2) n � Example: (cont’d) 3 n +1 n =0 x = − 5 : The power series becomes ∞ ∞ ∞ n ( − 5 + 2) n n · ( − 3) n n · ( − 1) n � � � = = , 3 n +1 3 n +1 3 n =0 n =0 n =0 which diverges since the terms do not converge to 0. x = 1 : The power series becomes ∞ ∞ ∞ n (1 + 2) n n · 3 n n � � � = 3 n +1 = 3 , 3 n +1 n =0 n =0 n =0 which diverges since the terms do not converge to 0. So the radius of convergence is R = 3 and the interval of convergence is ( − 5 , 1) (or − 5 < x < 1). Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series What If the Power Series Is Geometric? ∞ � ar k , If a power series is a geometric series, i.e. of the form k =0 then the general procedure used to compute the radius and interval of convergence is unnecessary. Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series What If the Power Series Is Geometric? ∞ � ar k , If a power series is a geometric series, i.e. of the form k =0 then the general procedure used to compute the radius and interval of convergence is unnecessary. We can find the interval of convergence simply by recalling ∞ ar k converges (absolutely!) if and only if | r | < 1 . � that k =0 Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II
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