1. ## quick limit

I have to figure out this limit:
Let (an) be a sequence whose limit as n approaches infinity is 1. Suppose that an does not equal 1 for all n in the natural numbers. For any fixed k in the natural numbers, find the limit as n approaches infinity of
an + (an)^2 + (an)^3 + ..... + (an)^k - k
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an - 1.

I know the limit is 1+2+3+.........+k. Yet I can't solve it algebraically. It seems so many of the individual limits go to infinity. Can someone show me how to do this? Thanks. I really appreciate it.

2. $\displaystyle\lim_{n\to\infty}\frac{a_n+a_n^2+a_n^ 3+\ldots+a_n^k-k}{a_n-1}=$
$\displaystyle=\lim_{n\to\infty}\frac{a_n-1}{a_n-1}+\lim_{n\to\infty}\frac{a_n^2-1}{a_n-1}+\lim_{n\to\infty}\frac{a_n^3-1}{a_n-1}+\ldots+\lim_{n\to\infty}\frac{a_n^k-1}{a_n-1}=$
$\displaystyle=1+\lim_{n\to\infty}(a_n+1)+\lim_{n\t o\infty}(a_n^2+a_n+1)+\ldots+\lim_{n\to\infty}(a_n ^{k-1}+a_n^{k-2}+\ldots+a_n+1)=$
$\displaystyle=1+2+3+\ldots+k=\frac{k(k+1)}{2}$