[Beatmania]

First solution

$$(1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^3})...<(1+\frac{1}{1\cdot3})(1+\frac{1}{2\cdot4})(1+\frac{1}{3\cdot5})...=(\frac {2^2}{1\cdot3})(\frac{3
^2}{2\cdot4})(\frac{4^2}{3\cdot5})...=2$$

Second Solution

$$(1+\frac{1}{3})(1+\frac{1}{3^2})(1+\frac{1}{3^3})...<(2^\frac{1}{2})(2^\frac{1}{4})(2^\frac{1}{8})...=2^{\frac{1}{2}+\frac{1}{ 4}+\frac{1}{8}+...}=2$$