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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:

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 
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 

Thank you in advance for your help.

Date: 10/31/96 at 11:52:26
From: Doctor Ceeks
Subject: Calculus Problem Using Taylor Series

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
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Associated Topics:
High School Calculus

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