Date: Sep 6, 1999 2:59 PM
Author: Peter Percival
Subject: Re: True/False?

If the range of t is bounded above and below then there is such a

polynomial by the Weierstrass Approximation Theorem. If t is allowed to

be indefinitely large, positive or negative, then there is not for the

reason you note.

neenag@cableol.co.uk wrote:

> True or False? :

>

> There exists a polynomial P such that:

>

> | P(t) - cos(t) | <= 10^-6.

>

> |t|, meaning the absolute value of t.

>

> I said false. Because P(t) could be very large. Then cos(t) is

> comparatively small. Then the answer is a large positive number,

> greater than 10^-6.

>

> Although then I started to think along the lines of the Taylor series,

> because

>

> cos(t) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

>

> and this is valid for all values of t, so perhaps there could be a

> polynomial (I'm not sure what that would be).

>

> So on the other hand it could also be true. Which is the correct

> answer?

>

> Neenag@cableol.co.uk