Q:

Show, that the sequence $ a_n = \left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{9}\right)\cdot\ldots\cdot\left(1+\frac{1}{3^n}\right) $ converges.

I need to show, that the following sequence converges.
I think I can somehow do it by Riemann Integral, but I cannot figure out a way to extract $ \frac{1}{n} $ from it. I also cannot find two sequences which could let me show that it converges by the sandwich theorem. How to approach it?
$$ a_n = \left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{9}\right)\cdot\ldots\cdot\left(1+\frac{1}{3^n}\right) $$

A:

If you aplied logarithm you get that $\log{a_n} = \sum_{n=1}^{\infty} \log{(1+\frac{1}{3^n})}$. Now using that $\log{x} \leq x-1$ for $x>0$ you get that $\log{a_n} \leq 
 \sum_{n=1}^{\infty} \frac{1}{3^n}$ that is a geometric series that converges, so as $\log{a_n}$ converges, then $a_n$ converges.

