You are correct, the limit is infinite. One approach to proving it is to show that the reciprocals go to zero. So think about the related sequence

a_n = 2009n/n!.

This is a sequence of positive values, so it is bounded below by zero. Also note that for all n = 2009 that

a_(n+1) - a_n = 2009n [2009/(n+1)! - (n+1)/(n + 1)!] = 2009n/(n + 1)! (2009 - (n + 1)) < 0.

The last inequality holds because n = 2009.

So for all n = 2009, a_n is a decreasing sequence bounded below by zero. It follows that a_n ->0 as n-> 8.

Your sequence is the reciprocals. So your sequence has limit 8 as n->8.

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