(adsbygoogle = window.adsbygoogle || []).push({}); Taylor formula. Need help!!

we have f(a+h), where a-is a point, and h is a very small term, h->0. And we have the formula to evaluate the function y=f(x), around the point a, which is

f(a+h)=f(a)+f'(a)h+o(h) --------(*)

however when we want to take in consideration o(h) this formula does not work, so we want to come up with sth more appropriate. So we want to write it in terms of the value of a polynom of the form

P_n(h)=b_1+b_2h+b_3h^2+.....+b_n h^n,where b-s are coeficients, that are not depended on h.

So it also says that when h->0, than

f(a+h)=P_n(h)+o(h^n)

When we have n=1, than the value of the polynom is f(a)+f'(a)h, according to (*). What i do not understand comes right here, when we want to generalize this for n.

It says that if the n-th derivative of f(x) exists, in particular f^(n) (a), than the polynom can be writen like this, or actually the polynom is:

P_n(h)=f(a)+[f'(a)h/1!]+[f"(a)h^2/2!]+.....+[(f^(n) (a)/n!)h^n],

Now my question is how did we come to this? In particular wher did we, or how did we derive this coefficients like f'(a)/1! , f"(a)/2! etc???

I would really appreciate anyones help!!

For two days i am stuck with this, i cannot just fathom it.

Thnx in return

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# Taylor formula. Need help

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