The trigonometric values also has a infinite series which I had learnt from the topics of Limits
Sin x = x - x³/ 3! + x⁵/ 5!.......
There are formulas like these for other trigonometric functions too. But here's where I don't understand. If we substitute 'π' instead of 'x' , then by using this formula , we get sin π not zero when it is zero. Similarly for all sin values of nπ ,where n ∈ Z get into the same trap.
sinπ = π - π³/3! + π⁵/5!...
0 = π - π³/3! + π⁵/5! ...
L.H.S ≠ R.H.S
This shouldn't happen, right ?
Where did I go wrong, then ?
Compute the partial sums $$S_p=\sum_{n=0}^p (-1)^n \frac{\pi^{2n+1}}{(2n+1)!}$$ For $p=4$, $S_4=0.00692527$, for $p=5$, $S_5=-0.00044516$, for $p=6$, $S_6=0.0000211426$.