Final Practice T1 Q10 - Looking for a detailed worked solution with theoretical explanation

This question has several parts. Please scroll down and try to answer all sub-questions until you get to "END OF THIS QUESTION". This question is designed so that you can answer each sub-question independently. For example, you can attempt b) without solving a), or you can solve c) without solving a) and b). Note that you must enter exact answers, for example, for 1/3 do not enter a decimal approximation such as 0.3333333333. For negative denominators you must use brackets, for example, -3/2 or 3/ (-2) but not 3/-2. Enter pi and exp (1) for 7 and the Euler number e. It is given that an infinitely many times differentiable function f (@) has Taylor series f (x) = 7 x2 + x3+ {terms of higher degree} (1) 16 valid in an interval containing zero. (a) Find f (0) = f' (0) = f" (0) = f'll (0) = (b) The coefficients in the right hand side of (1) are given by the sequence an = 2.271 , n _ 0. Find the radius of convergence R of the series. R = (c) Use the infinite series on the right hand side of (1) to compute f (1/9). If you find that the series diverges for x = 1/9, then type divergent in the box. f (1/9) = (d) Use the infinite series on the right hand side of (1) to compute f (-1/9). If you find that the series diverges for x = -1/9, then type divergent in the box. f (-1/9) = END OF THIS