Re: Power series for Cos x converge for all x?

From: Virgil (ITSnetNOTcom#virgil_at_COMCAST.com)
Date: 10/20/04 

Date: Tue, 19 Oct 2004 22:34:55 -0600

In article <cl4moc$bcu$1@nntp.itservices.ubc.ca>,
 israel@math.ubc.ca (Robert Israel) wrote:

> In article
> <ITSnetNOTcom#virgil-BC817B.18344819102004@comcast.dca.giganews.com>,
> Virgil <ITSnetNOTcom#virgil@COMCAST.com> wrote:
> >In article <cl4b66$en4$1@news5.svr.pol.co.uk>,
> > "fellow" <Jameson@god.com> wrote:
>
> >> Over what range of values of x does the standard power series for Cos x
> >> converge? In two text books it says for ALL x would you believe.
>
> >It does converge for all x, real or complex, but unless x is reasonably
> >close to zero, it may require an awful lot of terms to get a good
> >approximation.
>
> >For real x, the error in stopping at the term of degree 2*n will
> >eventually get to be less that term, x^(2*n)/(2*n)!, but, IIRC, this is
> >only guaranteed when n > |x/2|.
>
> YDRC.
> cos(x) is always between the sum to the x^(2*n-2) term and the
> sum to the x^(2*n) term, so the error will always be less than the
> absolute value of the last term. To prove, start with
>
> x - sin(x) >= 0 (for x >= 0)
>
> and integrate from 0 to x repeatedly.
>
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada

Thanks. It has been a long time since I last worked on it.

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