Test for convergence (n!)^3/(3n)! ratio test limit expand factorials and cancel to compute the limit
Test for convergence (n!)^3/(3n)! ratio test limit expand factorials and cancel to compute the limit.
We apply the ratio test to the series (n!)^3/(3n)!. We take the large n limit of the ratio of the n+1 term to the nth term and obtain an expression with many factorials in it. The trick to computing limits with factorials is to split off the leading terms of the factorials until you find a new factorial that will cancel.
We cancel several terms and arrive at the large n limit of a rational expression with a cubic polynomial in the numerator and denominator. There is no reason to keep more than the highest power of n in the numerator and denominator since lower powers of n are negligible in the limit as n goes to infinity (but if you prefer a more formal method of taking the limit, I describe how to do it).
Keeping the highest power of n and simplifying the limit, we arrive at a limit of 1/27 and because this is less than 1, the series converges by the ratio test.
