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This is how I think of it too, and it makes some part of me want to believe that means I have an intuitive grasp do what’s going on. Except I don’t think that’s

This is how I think of it too, and it makes some part of me want to believe that means I have an intuitive grasp do what’s going on. Except I don’t think that’s how math works. Rigor is necessary because an intuitive hand-wavey understanding is often wrong. I think there’s just as convincing a hand-wave available that dismisses the idea. 1) We have one point of the function. 2) We know one point of its derivative. We still don’t know any other points of the function. 3) We know one point of its second derivative. We still don’t know any other points of the function or its derivatives. ... N) Same deal. N+1) Same deal. ... Inf) Now we know all the points in the function and also all the points in all the derivatives. It looks absurd like this. Not to mention that we start with a countable set of points and derive a continuum from each.

lawrenceyan 7y ago
It's more like at each step from n -> n+1, our understanding grows by some amount. The way you write it seems to imply that no information is gained until some arbitrary point when n is extremely large.
Sharlin 7y ago
The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single point by definition tells us something about the neighborhood of that point. Mentally expanding the series terms to their limit forms helps see that it's not just a random polynomial. I guess the real "miracle" is that for a large class of useful functions those ugly limits evaluate to simple forms.
eridius 7y ago
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.

lawrenceyan 7y ago

It's more like at each step from n -> n+1, our understanding grows by some amount. The way you write it seems to imply that no information is gained until some arbitrary point when n is extremely large.

Sharlin 7y ago

The moment we start taking derivatives we leave the world of countable sets and move to limits and epsilon neighborhoods. Evaluating a derivative at a single point by definition tells us something about the neighborhood of that point. Mentally expanding the series terms to their limit forms helps see that it's not just a random polynomial. I guess the real "miracle" is that for a large class of useful functions those ugly limits evaluate to simple forms.

eridius 7y ago

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.