Hello again,
I decided to test the highlighted solution just for the fun of it. Here's what came of it.
For the following parameters:
g=9.8 meters/sec^2, L=1 meter, ang(0)=0 rads, ang'(0)=0.3 rads/second
where L is the length of the 'arm', ang(0) is the angle when the mass passes through x=0, and ang'(0) is the rate when the mass passes that same point. The equation for the pendulum was reduced to a set of two non linear ODE's and then calculated using a numerical routine.
The results were:
analytic max angle: 0.0958681928991643
max angle : 0.0958681928991863
analytic period: 2.0082434419777684
actual period: 2.0082436000490524
Last derivative: -3.705899254203889e-08
run time: 71.3 seconds
where
'analytic max angle' is the precise angle calculated using conservation of mechanical energy,
'max angle' is the angle calculated using a Runge-Kutta Fehlberg fourth/fifth numerical ODE solution routine,
'analytic period' is the precise period calculated using the highlighted formula up to N=22 (which causes high accuracy convergence),
'actual period' is the actual period tested,
'Last derivative' is the last derivative at the farthest swing angle to one side (would be 0.0 in an analytical calculation),
'run time' is simply the time it took to run the algorithm to completion.
Looking over the results we can see they do agree within a reasonable degree, especially the max angle calculations where the numerical calculation came out very close to the theoretical calculation. The actual period varies a bit more because the algorithm works by using a small increment and it can only be made just so small or it takes too long to complete, so the period accuracy does suffer, but it's close enough anyway and tests show that as the increment is reduced the actual period matches the analytic period closer and closer. The Last derivative would be zero if we did an exact calculation because the pendulum comes to a complete stop at the end of one swing to one side.
It took over a minute to calculate because i used such a small step time at an attempt to get decent accuracy