Let $b\_1,b\_2,\dots$ be positive real numbers such that 
$$s\_1<\infty\quad\text{and}\quad z\_1<\infty, 
$$
where 
$$s\_k:=\sum\_{j=k}^\infty b\_j\quad\text{and}\quad z\_k:=\sum\_{j=k}^\infty\frac{b\_j}{\sqrt{s\_j}} 
$$
for natural $k$. Does it then necessarily follow that 
$$\limsup\_{k\to\infty}\frac{z\_k}{\sqrt{s\_k}}<\infty\,\text{?}
$$
(One may note here that $z\_k\ge\sqrt{s\_k}$ for all $k$, whence $\liminf\_{k\to\infty}\frac{z\_k}{\sqrt{s\_k}}\ge1$.) 




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(This question is a "discretized" version of a problem considered in [this answer](https://mathoverflow.net/a/348149/36721).)



Comparing the tails of two related convergent series
In fact $z\_k$ (or better any partial sum for it) is a lower Riemann sum for the function ${1\over \sqrt{x}}$ wrto the infinitesimal decreasing sequence $s\_k>s\_{k+1}>\dots s\_{m} $, so $z\_k< \int\_0^{s\_k}{1\over \sqrt{x}}dx=2\sqrt{s\_k}$, and the constant $2$ is sharp.


