i know the limit should be 0 because the the series in the numerator is larger the the one in the denominator for every n
but i don't know how to start on proving it
help?
I believe you mean the denominator is larger than the numerator; therefore, the limit is 0. Wouldn't proving (and then pointing out that the product of two larger numbers is larger than the product of two smaller numbers) suffice?
You can start writing the limit as...
(1)
Now the limit (1) is called infinite product and a basic criterion exstablishes that if the infinite product is written in the form...
(2)
... where all the , then if the series...
(3)
... converges, then the infinite product (2) converges and if the (3) diverges, then the infinite product (2) tends to infinity if in (2) the sign is '+' and the infinite product tends to 0 if the sign in (2) is '-'. In Your case the series...
(4)
... diverges and the sign is '-', so that...
Kind regards