# Expression of the partial sum of a sequence

I was working on the following sequence $$u_n=\sum_{k=0}^{n}\frac{1}{q_0 \times q_1 \times \dots \times q_k}$$ where $$\left(q_n\right)$$ is an increasing sequence with $$q_0 \geq 2$$. I've shown that $$\left(u_n\right)$$ converges.

By trying $$q_n=2^{n+1}$$ for example, we would have $$u_n=\sum_{k=0}^{n}\frac{1}{2 \times 2^2 \times 2^{k+1}}=\sum_{k=0}^{n}\frac{1}{2^{1+2+\dots+\left(k+1\right)}}=\sum_{k=0}^{n}\frac{1}{2^{\left(k+1\right)\left(k+2\right)/2}}$$

Is there a way to compute this for all $$n$$, or to find its limit, apparently it approaches $$0,641633$$.

You have $$u_n=\frac{\left\lfloor c\,\, 2^{\frac{1}{2} (n+1) (n+2)}\right\rfloor } {2^{\frac{1}{2} (n+1) (n+2)} }$$ where $$c$$ is given in $$A190405$$ in $$OEIS$$ as already commented by @player3236.
If you want to make $$c$$ rational
$$c \sim \frac{436459}{680232}$$ is in error of $$6.23 \times 10^{-13}$$