Convergence of a series which is similar to a geometric series

Question

For all $k\in\mathbb{N}$, let $0<q_k\leq 1$. Assume that $\prod_{k=0}^\infty q_k=0$. Can we prove or disprove by counterexample that the series \begin{equation} \sum_{i=0}^\infty\prod_{k=0}^iq_k \end{equation} is convergent?

I understand that if for all $k\in\mathbb{N}$, there exists a $q$ such that $0<q_k\leq q<1$, then the series is convergent by comparison to $\sum_{i=0}^\infty q^i$.

Answer 1

Clearly not. Say $p_i=\prod_{k=0}^i q_k$. Then given any sequence $a_i$ with $0<a_i\le 1$, $a_{i+1}\le a_i$ and $a_i\to0$ there exist $q_k$ such that $p_i=a_i$.

Answer 2

A disproof by counterexample is $q_k = \frac{k+1}{k+2}$. The partial products form $\prod_{k=0}^nq_k = \frac{1}{n+2}$.

The sum is the diverging harmonic series, but $\frac1{n+2}$ converges to zero.