To determine The Taylor series about x 0 = 2 for the function f ( x ) = 1 1 − x and obtain the radius of convergence of the series. Explanation Formula used: If f has a power series expansion at x 0 , f ( x ) = ∑ n = 0 ∞ f ( n ) ( x 0 ) n ! ( x − x 0 ) n , f ( x ) = f ( x 0 ) + f ′ ( x 0 ) 1 ! ( x − x 0 ) + f ″ ( a ) 2 ! ( x − x 0 ) 2 + f ‴ ( a ) 3 ! ( x − x 0 ) 3 + ⋅ ⋅ ⋅ Ratio test: If lim n → ∞ | a n + 1 a n | = L < 1 , then the series ∑ n = 1 ∞ a n is absolutely convergent. Calculation: Consider the function f ( x ) = 1 1 − x centered at x 0 = 2 . Obtain the nonzero terms of the series as listed below. n f n ( x ) f n ( 2 ) 0 f ( x ) = 1 1 − x f ( 2 ) = 1 1 − 2 = − 1 1 f ′ ( x ) = ( 1 1 − x ) ′ = − ( − 1 ) ( 1 − x ) 2 = 1 ( 1 − x ) 2 f ′ ( 2 ) = 1 ( 1 − 2 ) 2 = 1 ( − 1 ) 2 = 1 2 f ″ ( x ) = ( 1 1 − x ) ′ ′ = ( 1 ( 1 − x ) 2 ) ′ = 2 ( 1 − x ) 3 f ″ ( 2 ) = 2 ( 1 − 2 ) 3 = 2 − 1 = − 2 3 f ‴ ( x ) = ( 1 1 − x ) ′ ′ ′ = ( 1 ( 1 − x ) 2 ) ′ ′ = ( 2 ( 1 − x ) 3 ) ′ = 6 ( 1 − x ) 4 f ‴ ( 2 ) = 6 ( 1 − 2 ) 4 = 6 ( − 1 ) 4 = 6 By the above formula, f ( x ) = ∑ n = 0 ∞ f ( n ) ( 2 ) n ! ( x − 2 ) n . Now substitute the values in f ( x ) = ∑ n = 0 ∞ f ( n ) ( 2 ) n ! ( x − 2 ) n as follows

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Chapter 5.1, Problem 16P

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The Taylor series about x0=2 for the function f(x)=11−x and obtain the radius of convergence of the series.

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Chapter 5.1, Problem 15P

Chapter 5.1, Problem 17P

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