The Ratio Test will usually cause a lot of cancellation in these cases; cancellation which will rid you of most, if not all, of the factorial part.
For instance, look at:
n^n/(n+1)^n = [n/(n+1)]^n --> 1/e as n->infinity
The reason is that [(n+1)/n)]^n=[1+1/n]^n -> e as n->infinity. This is by definition of the exponential.
Hence, according to the Ratio Test the original series is convergent.
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