This sequence converges to 1.
Proof:
If n > 1, then
> 1. Therefore,
we can find numbers an > 0 such that
=
1 + an for each n > 1
Hence, we can raise both sides to the power n and use the Binomial theorem:
In particular, since all terms are positive, we obtain
Solving this for an we obtain
0 
an
But that implies that an converges to zero
as n approaches to infinity, which means, by the definition of
an that
converges to 1 as n goes to infinity. That is what we wanted to prove.
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