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