Any faulty logic?

Prove: if a>1, (a^n)/n --> infinity.

Since a>1 we can write:

a = [(1 + k)]/ [n^(1/n)];

n*a^n = (1 + k)^n;

n*a^n =1 + nk + [n(n -1 )(k^2)]/2!+.....;

So, a^n =1/n + k +.....;

>1/n + k;

>M, for M>0 for say 1/n > M.

Is there a problem here since n < 1/M? Or, does this still imply n > M, so it doesn't matter?

Thanks.