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SKYBLUE1015
SKYBLUE1015
29.03.2021 • 
Mathematics

The function f has a Taylor series about x equals 1 that converges to f (x )for all x in the interval of convergence. It is known that f (1 )equals 1, f apostrophe (1 )equals minus 1 half, and the derivative of f at x equals 1 is given by f to the power of (n )end exponent (1 )equals (minus 1 )to the power of n fraction numerator (n minus 1 )factorial over denominator 2 to the power of n end fraction for n greater or equal than 2. Write the first four nonzero terms and the general term of the Taylor series for f about x equals 1. The Taylor series for f about x equals 1 has a radius of convergence of 2. Find the interval of convergence. Show the work that leads to your answer. The Taylor series for f about x equals 1 can be used to represent f (1.2 )as an alternating series. Use the first three nonzero terms of the alternating series to approximate f (1.2 ). Show that the approximation found in part c is within 0.001 of the exact value of f (1.2 ).

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