Ratio Test inconclusive with factorial

[micast87]

Homework Statement

$$\sum(n-1)!/(n+2)!$$

2. The attempt at a solution

I tried the ratio test and came up with the lim$$_{}n\rightarrow\infty$$ n/(n+3) = 1 which gives no information on convergence or divergence. I'm trying to find absolute or conditional convergence so what else can I do?

 

[physicsman2]

You can first simplify that sum into 1/[(n+2)(n+1)(n)] = 1/[n^3 +3n^2 + 2n].

Then do comparison test using another sum: 1/n^3,

Since the latter is greater than the former and it converges, therefore the original series converges. For absolute convergence, just remember to take the absolute value of the original series.