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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:	martin.krenn@dasa.dbmail.dbp.de

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?

```

```
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!
```

```
```
Associated Topics:
High School Calculus

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