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**1. an= [(2^n)(n!) + 1]/(n^n)**

At first glance, I thought it would be a convergent sequence because of the (n^n) in the denominator. But after trying and not being able to show its convergence, I compared it to (2/n)^n with the squeeze theorem. (Basically inf > an > (2/n)^n)

I set the limit as n approaches inf, took the natural log and put 1/2n as the denominator. I used the L'Hopital method and got to lim e^((-2/n^2)(n/2)/(-1/n^2)), which canceled out to lim e^n.

I concluded that the sequence diverges. Did I do this correctly? This problem's been bugging me for two days now. Guidance please!