Can the Infinite Product $\prod_{n=1}^\infty \frac{a_n}{a_n + 1}$ Converge or Diverge to Zero?

Question

Can the infinite product $\prod_{n=1}^\infty \frac{a_n}{a_n + 1}$ converge or diverge to zero when the sequence $a_n$ is monotonically increasing? I conjecture that this is impossible given that $\lim_{n\to\infty}\frac{a_n}{a_n+1}$ by the monotonic increase of the $a_n$. However, I don't know of a rigorous proof or disproof of this, and analyzing convergence of infinite series and infinite products is not something I am very familiar with. If someone could provide a proof or disproof of this conjecture (preferably with a reference to a known theorem), or explain what additional restrictions need to be imposed on $a_n$ for the conjecture to hold, I would be very appreciative!

Answer 1

Let $q_n=\dfrac{1}{1+a_n}$ be such that $0\lt q_n\lt 1,$ then by the elementary theory of infinite products $\prod_{n=1}^\infty (1-q_n)$ converges to a non-zero number if and only if $\sum_{n=1}^{\infty}q_n$ converges.
This clearly gives you a criteria to find the convergence to a non-zero number.

For example: if you set $a_n$ to be the $n$th prime (as mentioned in your comment), then $\sum_{n=1}^{\infty}q_n$ diverges and therefore ....

Answer 2

Every term is less than $1$, so it converges to a constant less than 1.

Take $a_n=n$, and the product converge to $0$.

Take $a_n=2^n$, and the product converges, but not to $0$. This can be obtained by taking the logarithm and using the inequality $\ln(1+x)<x$.