1) We have one point of the function. 2) We know one point of its derivative. This means we know 2 more points of the function (one step in either direction).

1) We have one point of the function. 2) We know one point of its derivative. This means we know 2 more points of the function (one step in either direction). 3) We know one point of its second derivative. This means we know 2 more points of the derivative, which in turn gives us 2 more points of the function. etc. Or at least, that's the impression I get from this thread, since I wasn't familiar with the series before that.