R-K weights?

A quick question for you guys, im going over past papers for an upcoming exam and a question on one of the PP is something along the lines of

"Show that this RK Order (whatever) agrees with taylors series" which i can do but as a tag along, it asks for an explination of what the k-values are.

ie

k1=f(
k2=...
k3=......
etc

Anyone who's seen it must know what i mean! Anyways, its not asking for an essay, but a shortl explination of what it is.

I think that they estimate different values of y(n+1) at various points using taylor series method of increasing order( and thus, accuracy?) , and then the final function takes some kind of weighted average of them. It makes sense in my head (but things always do !) but im wondering if anyone can explain it a little bit more eloquently?

What would you guys put?

Thanks

Assuming that the ODE is of the form

$\frac{dy}{dx} = f(x)$ ,

then I would put:

They are successively higher order, in terms of the Taylor series expansion, estimations of the integral

$\int_{x_n}^{x_{n+1}} f(x) \, \mathrm{d}x = y_{n+1} -y_{n}$

that when averaged, with suitable weighting, optimise the overall order achieved of the estimation of this integral.

Well that's just off the top of my head but I think that's all you need to say in a nutshell - at least without writing an essay!

So you had the right idea with your answer but were only wrong in that you thought they were an estimate of $y_{n+1}$ when in actual fact they are estimates of $(y_{n+1}-y_{n})$ .

Hope that was of some help!

Thaks for that man!