My calculus book explained how to form macluarin polynomials but said nothing on why they work. Over the past couple days I've been turning it over in my head trying to figure it out. I haven't found a proof but its begining to make sense. I thought it might make an intresting discussion for anyone who has never seen the proof or is not advanced enough to understand it yet.

Say the nth derivative of a function evaluated at zero is 5. If it were a polynomial with a limited number of terms, the nth derivative could be 5 itself. Lets integrate to find the (n - 1)th derivative 5x + c.  If the (n - 1)th derivative evaluated at zero were 7, then c would be 7.
OOH THIS IS FUN! Lets do it again! Integrate to find the (n - 2)th derivative, we get 5/2 x^2 + 7x + C. If at zero the (n - 2)th derivative is 14, then C would be 14.  get  So we've found the (n - 2)th derivative.

Now lets assume the n we spoke of had a value of 3, then the (n - 3)th derivative would be the 0th derivative, or the function itself.  So lets integrate the (n - 2)th derivative, to get 5/6 x^3 + 7/2 x^2 + 14x + C, if at zero the function had a value of 17 then C would be 17. So the function would look like
5/2! x^3 + 7/2 x^2 + 14x + 17 if it were in fact a polynomial with n terms.

Ok, now lets look back at what happened. Instead of saying (n - 1), (n - 2) or (n - 3) for each scenario lets just say u for whatever derivative we're working with at the time. The u'th derivative evaluated at zero was always some consant "c" (where u is some number between n and 0) this term is integrated u times and appears in the final function. Anyone familiar with calculus can see that the this term would end up being ( c x^u )/ u!.

So the function of x would be Summation of the u'th derivative at (0) multiplied by (x^u) / u! from u = infinity (ideally) to u = 0. We could reverse the order of the sum (which won't effect the answer) and sum from u = 0 to u = infinity, we would end up with:

f(0)/0! + f'(0)x/1! + f''(0)/2! + f'''(0)/3! ......

Well what do you know? The macluarin polynomial! Like I said this is not a proof. We made a bunch of assumptions, assuming the function was a polynomial when it may not be. We just used the clues to "mold" a polynomial with similar characteristics, (or exact when evaluated at 0) but I suppose the numerous stipulations tie the curve down to various places, and kind of leave little room for the function to stray far from the other on the inbetween points. Also we assumed the nth derivatve was a constant when it may not be at all. The nth derivative could have been 5 cos(x) However, if the n is very very large then this mistake should have little effect on the final function. Why? Because the limit of a x^n/n! as n approaches infinity is in fact zero. (I forget how you prove this but I remember doing it. Hmm... gotta review) Anyway, that doesn't prove it converges but it would prove it diverges if if the limit was not zero. So its at least consistant.

Anyways, this doesn't fully explain or proove why they work, but I think it gives you a pretty good idea of HOW they work. Like peeking under the hood of a car. Doesn't reveal everything but helps you understand it at least on a basic level. Whoever this Mr. Maclaurin was, he was a complete and utter genius who could probably move things with his mind.

Last edited by mikau (2006-07-01 09:40:48)