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.