Solve the Taylor Series. 1. (a) Use the root test to find the interval of convergence of-1)* に0 (b) Demonstrate that the above is the taylor series of f()- by writing a formula for f via taylor's theorem at α-0. That is write f(x)-P(z) + R(x) where P(r) is the nth order taylor polynomial centered at a point a and the remainder term R(x) = ((r - a)n+1 for some c between z and a where here a 0. Show that R()0 as n OO 1+1 : Σ rnor Σ(-1)krk for all z in the interval of (c) Above you have f(x)-__ ) 1 + r K! convergence you determined in part a. Integrate both sides of this equation and find the interval of convergence for the integrated series (d) Find a taylor series representation for In(1) in exactly the way you did for part b. After you find the taylor series find the interval of convergence 1. (a) Use the root test to find the interval of convergence of-1)* に0 (b) Demonstrate that the above is the taylor series of f()- by writing a formula for f via taylor's theorem at α-0. That is write f(x)-P(z) + R(x) where P(r) is the nth order taylor polynomial centered at a point a and the remainder term R(x) = ((r - a)n+1 for some c between z and a where here a 0. Show that R()0 as n OO 1+1 : Σ rnor Σ(-1)krk for all z in the interval of (c) Above you have f(x)-__ ) 1 + r K! convergence you determined in part a. Integrate both sides of this equation and find the interval of convergence for the integrated series (d) Find a taylor series representation for In(1) in exactly the way you did for part b. After you find the taylor series find the interval of convergence