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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 $0,641633$.

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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 $6.23 \times 10^{-13}$

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