provide a full handwritten solution for each question

1. A useful power series. Let! Jo(2) =[)"al 220 (n!)2 (a) Find the radius of convergence and the interval of convergence of the power series above (i.e., the set of values for which Jo(a) is defined). (b) What is the value of f(7) (0)? (c) What is the value of f(1009) (0)? Note: do not evaluate large factorials or powers in the above expressions. 2. Taylor series. (a) Find the Taylor polynomial of degree 3 centered at 1 of the function /(x) = e- (b) Use the binomial series to estimate the arclength of the graph of function /(a) = -, from the points (0, 0) to (1/2, 1/24). Use only the first three terms of the series. 3. Taylor series from graph. If the graph of f is Explain why each of the following series cannot be the Taylor series of f centered at the point given (a) 1.5 -0.75(2 - 1) + 1.5(2 - 1)3 -0.1(x - 1)* + ... (centered at a = 1) (b) 1.2 - 0.75(2 - 5) - 1.2(x - 5)2 -0.2(x -5) + ... (centered at a = 5)