>>13055251Consider a polynomial . It's easy to see that the k-th coefficient is given by , or equivalently that the k-th derivative evaluated at is . So for polynomials there's a clear and simple relationship between its coefficients and derivatives evaluated at zero. Knowing one thing is equivalent to knowing the other thing.
Now do the same thing for a general function . If it was a polynomial, its coefficients would be precisely . Even if is not polynomial, you can still consider a polynomial with these coefficients. Result is the Taylor polynomial. It's the "pretend that is a polynomial" polynomial.
