Explanation Theorem used: Taylor Series Representation of a Function: If a function f has derivatives of all orders in an open interval I = ( a − R , a + R ) ( R > 0 ) centered at a, then f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n if and only if lim n → ∞ R N ( x ) = 0 for all x in I, where R N ( x ) = f ( x ) − P N ( x ) . Calculation: The given series is f ( x ) = 1 x + 1 . Here, a = 0 . Find f ( x ) and f ( 0 ) as follows. f ( x ) = 1 x + 1 and f ( 0 ) = 1 . Find f ′ ( x ) and f ′ ( 0 ) as follows. f ′ ( x ) = − 1 2 ( x + 1 ) 3 2 f ′ ( 0 ) = − 1 2 ( 0 + 1 ) 3 2 = − 1 2 Find f ″ ( x ) and f ″ ( 0 ) as follows. f ″ ( x ) = − 1 2 ( − 3 2 ) ( x + 1 ) − 3 2 − 1 = 3 4 ( x + 1 ) 5 2 f ″ ( 0 ) = 3 4 ( 0 + 1 ) 5 2 = 3 4 Find f ‴ ( x ) and f ‴ ( 0 ) as follows. f ‴ ( x ) = 3 4 ( − 5 2 ) ( 1 − x ) − 5 2 − 1 = − 15 8 ( x + 1 ) 7 2 f ‴ ( 0 ) = − 15 8 ( 0 + 1 ) 7 2 = − 15 8 Similarly, find f ( n ) ( x ) and f ( n ) ( 0 ) as follows. f ( n ) ( x ) = ( − 1 ) n 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) 2 n ( x + 1 ) 1 2 + n and f ( n ) ( 0 ) = ( − 1 ) n 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) 2 n

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 11.5, Problem 31E

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To find: The Taylor series of the function f(x)=1x+1 at the indicated number x=0 and give its radius and interval of convergence.

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Chapter 11.5, Problem 30E

Chapter 11.5, Problem 32E

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Chapter 11 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 11.1 - Prob. 1CQ Ch. 11.1 - Prob. 2CQ Ch. 11.1 - Prob. 1E Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E

Ch. 11.1 - In Exercises 110, find the first three Taylor...Ch. 11.1 - Prob. 10E Ch. 11.1 - Prob. 11E Ch. 11.1 - Prob. 12E Ch. 11.1 - Prob. 13E Ch. 11.1 - Prob. 14E Ch. 11.1 - In Exercises 1120, find the nth Taylor polynomial...Ch. 11.1 - Prob. 16E Ch. 11.1 - Prob. 17E Ch. 11.1 - Prob. 18E Ch. 11.1 - Prob. 19E Ch. 11.1 - Prob. 20E Ch. 11.1 - Prob. 21E Ch. 11.1 - Prob. 22E Ch. 11.1 - Prob. 23E Ch. 11.1 - Prob. 24E Ch. 11.1 - Prob. 25E Ch. 11.1 - Prob. 26E Ch. 11.1 - Prob. 27E Ch. 11.1 - Prob. 28E Ch. 11.1 - Prob. 29E Ch. 11.1 - Prob. 30E Ch. 11.1 - Prob. 31E Ch. 11.1 - Prob. 32E Ch. 11.1 - Prob. 33E Ch. 11.1 - Prob. 34E Ch. 11.1 - Let f(x) = ex. a. 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The Taylor series of the function f ( x ) = 1 x + 1 at the indicated number x = 0 and give its radius and interval of convergence.

Solution Summary: The author explains the Taylor series of the function f(x)=1sqrtx+1 and its radius and interval of convergence.

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To find: The Taylor series of the function f(x)=1x+1 at the indicated number x=0 and give its radius and interval of convergence.

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Chapter 11 Solutions