I've gotten the definition that $$P_n$$ is a good approximation when :
$$ \lim_{x\to a} \frac{f(x)-P_n(x)}{{(x-a)}^n} = 0$$
But I am having a hard time understanding what that means. Also:
We have a formula of something along the lines of
$$P_n(x) = c_0 + c_1(x) + c_2x^2 +...$$
But I don't understand what the meaning is and how it is different from the definition above.
If I were to asked to write the Taylor Polynomial for $$f(x) =x^3 $$ at a = 0. How would I find the values of $$P_2(x), P_3(x), P_4(x) $$and so on. Would it be 0 in every case. Because thats what I am getting.
Notice that the constant function $0$ is a good first and second order approximation to $x^3$ because
$$\lim_{x \to 0} \frac{x^3 - 0}{x^n} = \lim_{x \to 0} x^{3 - n} = 0$$
provided that $n < 3$. If you consider any other approximation, such as $c_0 + c_1 x$, you would find that
$$\lim_{x \to 0}\frac{x^3 - c_0 - c_1 x}{x^n}$$
doesn't exist when $c_0$ and $c_1$ are both non-zero.
Of course this breaks down when $n = 3$, in which case $x^3$ is the best approximation (although a rather trivial one in this case... the Taylor series of a polynomial is just the polynomial itself). In that case, we have
$$\lim_{x \to 0}\frac{x^3 - x^3}{x^n} = 0$$
for all $n$.