3. Finding a formula for the inverse of a. function can be extremely hard even if we know the inverse exists. Miraculously, if the function is analytic then its inverse can be explicitly computed as a power series: Theorem. Let I be an open interval and let f be an analytic function on I. Fiat a E I and I) = f(a). IF Va: E I, f'($) 7E 0 THEN f'l exists and is also anaiytie on its domain. Moreover, for y near I), DO f'1(y) : 2 3(2)! - 5)\" 71:0 where cu : f'1(b) : a and. V\" E N\" '3\" : Hi. lei-:1 Km)\" You will assume1 this theorem to compute the inverse of f(a:) = mew as power series. (Try the usual approach to nding an inverse of f. You'll quickly see it's impossible.) (C) Let f"1 be the inverse of f on I. Compute f'1 as a power series expansion centered at 0 using the above theorem. Your answer should be a series written using sigma notation. (f) Let a: E I be the value such that 3351' = i. Approximate 53 Within an error less than 0.01. Your answer should be expressed as a fraction. Note that Z is in the interval convergence for the power series from (3c)