Why Does Adding the nth Derivative Increase a Function Approximation's Accuracy? I am currently taking calculus 3: sequences and series, and we've just started learning about Maclaurin and Taylor Series. I understand the concept behind them -- of these polynomials and derivatives of polynomials. 
However, I do not understand physically why, when we have a function approximation $g(x) \approx f(x)$, adding more and more derivatives of $f(x)$ increases the accuracy of $g(x)$ more and more.
If someone could point me to a resource or explain it in simple terms it would be much appreciated. Thank you!

Consider a polynomial function $P$ and let $x_0$ be a real number. Then $P(x_0)$ is the value that $P(x)$ takes at $x_0$ and, since polynomial functions are continuous, whe $x$ is close to $x_0$, then $P(x)$ is close to $P(x_0)$.
Now, consider the polynomial $P_1(x)$ of degree $1$ such that $P_1(x_0)=P(x_0)$ and that $P_1'(x_0)=P'(x_0)$. It turns out that $P_1(x)=P(x_0)+P'(x_0)(x-x_0)$. On the other hand, if you write $P(x)$ as polynomial in $x-x_0$ rather than in $x$, what you will get will be$$P(x)=P(x_0)+P'(x_0)(x-x_0)+\frac{P''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{P^{(n)}(x_0)}{n!}(x-x_0)^2.$$So, $P_1(x)$ is the best first degree approximation of this function near $x_0$, in the sense that$$\lim_{x\to x_0}\frac{P(x)-P_1(x)}{(x-x_0)^2}\tag1$$exists; if in $(1)$, $P_1(x)$ is replaced by any other first degree polynomial, that limit will not exist.
Now, if $P_2(x)=P(x)=P(x_0)+P'(x_0)(x-x_0)+\frac{P''(x_0)}{2!}(x-x_0)^2$, then $P_2(x)$ is the only second degree polynomial such that the limit$$\lim_{x\to x_0}\frac{P(x)-P_2(x)}{(x-x_0)^3}$$exists, and so on.
And since most functions behave locally like polynomials, approximating such a function $f$ by$$f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2!}(x-x_0)^2+\cdots+\frac{f^{(n)}(x_0)}{n!}(x-x_0)^2,$$will be, in general, a good idea.
