I have been trying to utilize the formula:
I simply cannot figure out how to determine the error when using
Instead of:
I have made a Taylor polynomial around 0 of grade 4, and I cannot find the deviation when using the Taylor polynomial instead of the original function.
The Taylor polynomial is:
And it was made of the Lorentz factor:
Let us call $f(t)$ the function $$f(t) = \frac{1}{\sqrt{1-t}} $$
The $k^{th}$ derivative function of $f()$ is equal to $$f^{k}(t) = \frac{\prod_{i=1}^k{(2i-1)}}{2^k} (1-t)^{-\frac{2k+1}{2}} $$ This last relation can for example be demonstrated by recurrence.
According to Taylor-Lagrange formula, there exists $\xi$ between $0$ and $t$ such that
$$f(t) = 1 + \frac{t}{2} + \frac{3}{8}t^2 + \frac{5}{16}(1-\xi)^{-7/2}\,t^3 = 1 + \frac{t}{2} + \frac{3}{8}t^2 + R(t)$$
It follows
$$0 < R(t) < \frac{5}{16}(1-t)^{-7/2}\,t^3 $$
Now, one just have to replace $t$ : $$t = x^2 = \left(\frac{v}{c}\right)^2$$