My 'proof' doesn't use the 'Taylor series theorem' but avoids tedious computation of derivatives...
Let's strart with the hypothesis that it exists the McLaurin expansion...
(1)
...and we want to compute the . Now, considering that is...
(2)
... setting in (2) we derive using (1) and a little of patience...
(3)
... and the identity (3) permits the computation of the ...
Kind regards