## Question

###### Advance Maths Problem 45Please donâ€™t answer compulsory I will givedislikeProblem 45: Lagrangeâ€™s identity (âœ“âœ“âœ“) 1987 Paper II If y = f(x), the inverse of f is given by Lagrangeâ€™s identity: f âˆ’1 (y) = y+ âˆ‘âˆž 1 1 n! d nâˆ’1 dy nâˆ’1 [ y âˆ’ f (y) ]n , when this seriesconverges. (i) Verify Lagrangeâ€™s identity when f (x) = ax. (ii)Show that one root of the equation x âˆ’ 1 4 x 3 = 3 4 is x = âˆ‘âˆž 0 32n+1 (3n)! n!(2n + 1)! 43n+1 . (â€ ) (iii) Find a solution for x, asa series in Î

Advance Maths Problem 45 Please donâ€™t answer compulsory I will give dislike Problem 45: Lagrangeâ€™s identity (âœ“âœ“âœ“) 1987 Paper II If y = f (x), the inverse of f is given by Lagrangeâ€™s identity: f âˆ’1 (y) = y + âˆ‘âˆž 1 1 n! d nâˆ’1 dy nâˆ’1 [ y âˆ’ f (y) ]n , when this series converges. (i) Verify Lagrangeâ€™s identity when f (x) = ax. (ii) Show that one root of the equation x âˆ’ 1 4 x 3 = 3 4 is x = âˆ‘âˆž 0 3 2n+1 (3n)! n!(2n + 1)! 43n+1 . (â€ ) (iii) Find a solution for x, as a series in Î», of the equation x = eÎ»x . [You may assume that the series in part (ii) converges and that the series in parts (i) and (iii) converge for suitable a and Î».] Comments This looks pretty frightening at first, because of the complicated and unfamiliar formula. However, its bark is worse than its bite. Once you have decided what you need to find the inverse of, you just substitute it into the formula and see what happens. Do not worry about the use of the word â€˜convergenceâ€™; this can be ignored. It is just included to satisfy the legal eagles who will point out that the series might not have a finite sum. In part (ii) you can, as it happens, solve the cubic by normal means (find one root by inspection, factorise and use the usual formula to solve the resulting quadratic equation). The root found by Lagrangeâ€™s equation is the one closest to zero. Equation (â€ ) turns out to be a very obscure way of writing a familiar quantity.29 Lagrange was one of the leading mathematicians of the 18th century; Napoleon referred to him as the â€˜lofty pyramid of the mathematical sciencesâ€™. He attacked a wide range of problems, from celestial mechanics to number theory. In the course of his investigation of the roots of polynomial equations, he discovered group theory (in particular, his eponymous theorem about the order of a subgroup dividing the order of the group), though the term â€˜groupâ€™ and the systematic theory had to wait until Galois and Abel in the first part of the 19th century. Lagrangeâ€™s formula, produced before the advent of the theory of integration in the complex plane, which allows a relatively straightforward derivation, testifies to his remarkable mathematical ability. It is practically forgotten now, but in its day it had a great impact. The applications given above give an idea how important it was, in the age before computers