How do you show that the infinite product of $(3n+1)/(3n+2)$ converges to zero without using the gamma function (as I can tell the gamma function solution has already been presented here)?
I've tried using a number of different strategies, but I still don't seem to be able to land on a good approximation.
When dealing with infinite product of positive real numbers, it is always a good ideal to consider the log-transform. Let $f:(-1,1)\rightarrow\mathbb{R}$ be defined by $f(x)=x+\ln(1-x)$, then $f'(x)=-\frac{x}{1-x}$. This shows that $f$ is strictly increasing on $(-1,0]$ and strictly decreasing on $[0,1)$. In particular, $f(x)\leq f(0)=0$ for all $x\in(-1,1)$. That is, $\ln(1-x)\leq-x$, for all $x\in(-1,1)$. We need this fact at later time.
Let $P_{n}=\prod_{k=1}^{n}\frac{3k+1}{3k+2}$, then we have \begin{eqnarray*} \ln P_{n} & = & \sum_{k=1}^{n}\ln\left(1-\frac{1}{3k+2}\right)\\ & \leq & \sum_{k=1}^{n}-\frac{1}{3k+2}. \end{eqnarray*} Therefore, $0\leq P_{n}\leq\exp\left(-\sum_{k=1}^{n}\frac{1}{3k+2}\right)$. Note that $-\sum_{k=1}^{n}\frac{1}{3k+2}\rightarrow-\infty$ as $n\rightarrow\infty$. Therefore $P_{n}\rightarrow0$ as $n\rightarrow\infty$.
