Find the series expansion of the solution of the equation ξ − e cos ξ = l and prove that it converges uniformly in l for |e| small enough. Let y = sin x = x − x3/3! + x5/5! + O(x7). Compute the...

Find the series expansion of the solution of the equation ξ − e cos ξ = l and prove that it converges uniformly in l for |e| small enough.

Let y = sin x = x − x3/3! + x5/5! + O(x7). Compute the expansion of x = X(y) up to terms of order O(y7). Verify the accuracy by comparing with the Taylor series expansion of x = arcsin y.