Does this product converge? 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. 
 A: $$P=\prod_{x=0}^{\infty}\frac{(30x+5)(30x+9)(30x+15)(30x+21)(30x+27)}{(30x+7)(30x+13)(30x+19)(30x+23)(30x+31)}$$
$$\frac{(30x+5)}{(30x+7)}  <1 $$
$$\frac{(30x+9)}{(30x+13)}  <1 $$
$$\vdots$$
$$P < 1$$
You are multiplying many numbers which are less than $1$ to the final result must be less than $1$.
A: 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 $P_N \to 0$. 
