Calculus Problem Using Taylor Series
From: mosaic account for www Subject: Calculus Problem Using Taylor Series I have a problem which I think can only be solved using iterative methods. If you know how to obtain an analytical solution, please respond to: email@example.com The problem is as follows: Equation: 1 + x*2^(x-1) Values must to obtained for 'x' which lead to a square of an integer. For example, if x = 5, then the result is 81 which is 9 squared. Another way of formulating the problem is: Equation: y^2 = 1 + x*2^(x-1) where y is an integer. If there is no solution, then is there any way to prove that the values obtained iteratively are the only possible solutions to the equation? Thank you in advance for your help.
Date: 10/31/96 at 11:52:26 From: Doctor Ceeks Subject: Calculus Problem Using Taylor Series Hi, Since f(x)= 1 + x*2^(x-1) is continuous and tends to infinity when x->infinity, and since f(0) = 1, there is always a solution for y^2 = f(x), for all integers y (although x may not be an integer). You might be able to find the Taylor series for the inverse function of f(x) using implicit differentiation. -Doctor Ceeks Check out our web site!
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.