Re: Cosine Product Limit

In article <Pine.BSI.4.58.0506111428380.3354@xxxxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:

> On Sat, 11 Jun 2005, Albert wrote:
>
> > I want to evaluate the following limit:
> >
> > lim[x->0](1-cos(x)*cos(2x)*...*cos(nx))/x^2
> >
> = lim(x->0) (1 - cos nx)/x^2
> + lim(x->0) (cos nx)(1 - cos x *..* cos (n-1)x)/x^2
> = n^2 / 2 + lim(x->0) (1 - cos x *..* cos (n-1)x)/x^2
> = induction mumble mumble
> = sum(j=1,n) j^2 / 2 = n(n + 1)(2n + 1)/6 * (1/2)
> = same answer as Stephen with same comment
>
> > How do I evaluate the cosine product: cos(x)*cos(2x)*...*cos(nx) using
> > the Taylor series?
> >
> Masochistically, unless you're expecting us to do it you vile sadist. ;-)

You only need it up to x^2 plus a O (x^4).

cos x = 1 - x^2 / 2 + O (x^4)
cos nx = 1 - n^2 x^2 / 2 + O (x^4) for any fixed n.

Product from cos x to cos nx:

1 - (sum k^2 for 1 <= k <= n) * x^2 / 2 + O (x^4)

Then the answer is the same.
.