I'll show you some highlights of what Euler did in his time since it is a delightful thing, though this is NOT a formal proof:
(1) , and substracting this from (1):
, and substracting this from (2) above:
Continue in this fashion and you'll get
Well, now just solve for above...
Ps If after seeing the above one doesn't crash down in deep love with mathematics then all is lost...
Ps of Ps BTW, what does the gamma function has to do with all this??