Does this product converge?

Question

Does this product converge? $$\prod_{x=0}^{\infty}\frac{(30x+5)(30x+9)(30x+15)(30x+21)(30x+27)}{(30x+7)(30x+13)(30x+19)(30x+23)(30x+31)}$$

I have tried solving this by hand, and it seems to get closer to zero.

Answer 1

$$P=\prod_{x=0}^{\infty}\frac{(30x+5)(30x+9)(30x+15)(30x+21)(30x+27)}{(30x+7)(30x+13)(30x+19)(30x+23)(30x+31)}$$

$$\frac{(30x+5)}{(30x+7)} <1 $$

$$\frac{(30x+9)}{(30x+13)} <1 $$

$$\vdots$$

$$P < 1$$

You are multiplying many numbers which are less than $1$ to the final result must be less than $1$.

Answer 2

Take the natural log and use the fact that $\ln(1-y) \le -y$ for $y > 0$ to get

$\ln P_N = \displaystyle \sum_{x = 0}^{N}\left[\ln\dfrac{30x+5}{30x+7} + \ln\dfrac{30x+9}{30x+13} + \ln\dfrac{30x+15}{30x+19} + \ln\dfrac{30x+21}{30x+23} + \ln\dfrac{30x+27}{30x+31}\right]$

$= \displaystyle \sum_{x = 0}^{N}\ln\left[1-\tfrac{2}{30x+7}\right] + \ln\left[1-\tfrac{4}{30x+13}\right] + \ln\left[1-\tfrac{4}{30x+19}\right] + \ln\left[1-\tfrac{2}{30x+23}\right] + \ln\left[1-\tfrac{4}{30x+31}\right]$

$\le \displaystyle -\sum_{x = 0}^{N}\left[\dfrac{2}{30x+7} + \dfrac{4}{30x+13} + \dfrac{4}{30x+19} + \dfrac{2}{30x+23} + \dfrac{4}{30x+31}\right] \to -\infty$ as $N \to \infty$ by comparison to the harmonic series.

Since, $\ln P_N \to -\infty$, we have that $P_N \to 0$.