Consider the following function f x x a 5 points Approximate f by a Taylor polynomial T3 x with the center x 1 Your answer must

Question

Consider the following function f x x a 5 points Approximate f by a Taylor polynomial T3 x with the center x 1 Your answer must be of the form co c x 1 c x 1 c x 1 b 5 points How good is the approximation on the small interval 0 9 x 1 1 Use the Taylor s Inequality to find an upper bound of the error R3 x f x T x c 5 points Find the Taylor series of f x with center at x 1 Use the summation notation whoch displays the n th term clearly Then determine the interval of convergence of this series d Extra Credit 3 points Show by using Taylor Ineqality that the absolute value of the n th remainder R x converges to 0 for every x in the interval 0 6 1 4 e Extra Credit 3 points Find a rational expression to which this series converges on its interval of convergence You must show your systematic procedure instead of simply making a wild guess Hint Try differentiation or integration

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