Find limit of sequence. Hint says that logarithm might be helpful.

Question

I have hard time with this one: $$ \lim_{n\rightarrow \infty}\frac{1*4*...*(3n+1)}{2*5*...*(3n+2)} $$ Hint says that using natural logarithm may be (but does not have to) be useful. Unfortunately this hint is not enough for me, so maybe someone could come up with something better.

Answer 1

HINT:

$$\log\left(\frac{\prod_{k=0}^{n}(3k+1)}{\prod_{0=1}^{n}(3k+2)}\right)=\sum_{k=0}^{n}\log\left(1-\frac{1}{3k+2}\right)$$

and for $z<1$

$$\frac{-z}{1-z}\le \log (1-z)\le-z$$

Then, compare the series with the harmonic series.