Derivation of Maclaurin expansion of Lorentz factor

Expand the Lorentz and you will be inlightened. Not just for the precision of afforded near the speed of light, but also insite to other series, such as a hyperbolic.

You can work in terms of the inverse or normal. I will derive in normal. Or when you are done, you can invert the individual iterated terms. That is easy.

You must use a substitution, otherwise the function would be in terms of derivatives of v. Derivatives of functions of v are not defined at zero.

I am first going to substitute a specific value, far from zero.
I am not really expecting much.
The value that I am going to choose is a velocity of the square root of two, of the velocity of light.
With this value, the square of beta represents one half of energy. And also, gamma represents an exact value of the square root of two - which is accurately known.

The first evaluation is with the computer simply evaluating the direct expression, in double precision. And it also agrees with the square root of two as found in tables (to 15 places). So this is "gospel"

Here are the iterated Laclaurin values.
Terrible approximations!
So, at least for this "far away" value, this kind of accuracy is useless.

Now for a close-in value to zero.
Great approximation!

Another value...
The same as a trig identity...

The value can be double checked
because the secant of .1 is also 1.00502091840046 to 15 places.

I had an alterior motive in looking at the Lorentz Factor: I wanted to also look at my computer.
My computer seems to have about 15 digits of accuracy in double precision. Double precision works with 18 native mantissa digits, outputting 15. Accuracy may start out with 15 digits; This is the most that can be expected! Accuracy immediately begins to suffer as soon as manipulations are started. It is sort of like damage to DNA strands. The more manipulations, the greater the damage, and the greater the loss of accuracy.

Also, one variable may be compromised in accuracy, and affect the accuracy of all the rest and the end solution. It is sort of like the weakest link thing. Order of operations can hurt the accuracy of any member, as head room is lost and rounding errors creep in.