I am wondering for $n > 1$ if the following series converges and if possible what it equates to.
$\sum _{m=0}^{\infty }\dfrac {\log\left( \dfrac {\left( n+m-1\right) !} {\left( n-1\right) !}\right) -m} {\prod _{t=n}^{t=n+m-1}\log t}$
I suspect it should sum to 1 as they are mutually exclusive probabilities, unless i made a mistake some where. I tried ratio test and wolfram alpha but did not have much success. Any help would be much appreciated.
Thanks in advance.