Proving a sum to be sublinear in growth Suppose that $\alpha \in (0,1)$. The goal is to prove that the following sum is of $o(T)$ (or, if possible, give a more accurate growth rate, e.g. $O(T^{1-\alpha})$ or something like that):
$$ \sum_{t=1}^T \prod_{i=0}^{\lfloor\sqrt{t}\rfloor-1}(1 - (t-i)^{-\alpha}).$$
It is also easy to see that the growth rate of the sum below is higher than the one above. So proving sub-linearity of the sum below also is enough:
$$ \sum_{t=1}^T (1 - t^{-\alpha})^{\sqrt{t}}.$$
 A: $\newcommand{\al}{\alpha}
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For $i=0,\dots,\lfloor\sqrt{t}\rfloor-1$, we have $t-i=ct$; here and in what follows $c$ denotes various positive quantities (which possibly differ from one another even in the same formula) bounded away from $0$ and $\infty$ over all 
\begin{equation}
\al\in(0,1),\quad T=1,2,\dots,\quad t=1,\dots,T,\quad i=0,\dots,\lfloor\sqrt{t}\rfloor-1;\tag{1} 
\end{equation} 
that is, there is a universal positive real constant $C$ such that for each of these instances of $c$ and for all $\al$, $T$, $t$, and $i$ as in (1) one has $1/C\le c\le C$. So, the sum in question is 
\begin{equation*}
 s_T:=\sum_{t=1}^T (1 - (ct)^{-\alpha})^{c\sqrt{t}}
 =\sum_{t=1}^T \exp\{-ct^{1/2-\al}\}. 
\end{equation*}
If $\al\ge1/2$, then $\exp\{-ct^{1/2-\al}\}=c$ and hence $s_T$ is $cT$ and not $o(T)$. 
If $\al<1/2$, then $\exp\{-ct^{1/2-\al}\}=o(1)$ as $t\to\infty$ and hence $s_T=o(T)$ as $T\to\infty$; in this case, we even have $s_T=O(1)$.  
