Approximating functions with Taylor polynomials and error bounds

To approximate a function with a Taylor Polynomial of n'th degree centred around $a=0$, use
$f(x) \approx f(a) + f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$

where $P_n (x) = f(a)+f'(a)(x-a)+\frac{f^{"}(a)(x-a)^2}{2!}+ \cdots + \frac{f^n (a)(x-a)^2}{n!}$ is the Taylor Polynomial.

To find the difference between the actual value and your approximated value, look for the error term, which is defined as
$R_n(x)=\frac{f^{n+1}(z)(x-a)^{n+1}}{(n+1)!}$

Note that adding your Taylor Polynomial with your error would give you the exact value of the function. In other words,
$f(x)=P_n(x)+R_n(x)$