Prove that this sequence converge. I am obliged to prove that this sequence:
$\large {a_n=(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{27})...(1+\frac{1}{3^n})}$
is convergent sequence.
I mean I was thinking about this and I know that $\large\lim_{n \to \infty} (1+\frac{1}{3^n})=1 $
From this I know that it will be probably convergent sequence but I know that it is not well written proof, and probably does not prove anything. I would be glad for any tips how to prove this. 

Take $\log$:
$$\log\lim_{n\to\infty} a_n = \lim_{n\to\infty}\log a_n = \lim_{n\to\infty}\sum_{k=1}^n\log(1+{1\over 2^k}) = \sum_{k=1}^\infty\log(1+{1\over 2^k});$$
now, as
$$\lim_{k\to\infty}{\log(1+{1\over 2^k})\over{1\over 2^k}}=1,$$
the behavior of $\sum\log(1+{1\over 2^k})$ is the same that the behavior of $\sum {1\over 2^k}$, and this series is convergent (geometric with reason < 1).
(Edited for greater clarity)
