<s>[INST] Finding the value of a sum using Riemann sum theorem 
Question: Find the value of $\sum_{i=1}^{n}(\frac{1}{n-i})^{c}$ for large $n$. 

\begin{align}
\sum_{i=1}^{n}(\frac{1}{n-i})^{c} 
& = \sum_{i=1}^{n}(\frac{1}{n})^{c}(\frac{1}{1-\frac{i}{n}})^{c}
\\ & = \frac{n}{n} \times \sum_{i=1}^{n}(\frac{1}{n})^{c}(\frac{1}{1-\frac{i}{n}})^{c} 
\\ & = n(\frac{1}{n})^{c}  \sum_{i=1}^{n}\frac{1}{n}(\frac{1}{1-\frac{i}{n}})^{c}  \qquad(1)
\end{align}
Let $f(x) = (\frac{1}{1-x})^{c}$, by using Riemann-sum theorem, we have
\begin{align}
\lim_{n\rightarrow \infty}\sum_{i=1}^{n}\frac{1}{n}(\frac{1}{1-\frac{i}{n}})^{c}
& = \int_{0}^{1} (\frac{1}{1-x})^{c} = A \qquad(2)
\end{align}
By using $(1)$ and $(2)$, for sufficently large $n$, we have
$$\bbox[5px,border:2px solid #C0A000]{\sum_{i=1}^{n}(\frac{1}{n-i})^{c} = A\times n(\frac{1}{n})^{c}}$$

The presented proof has a problem, $f(x)$ is not defined in the closed interval $[0,1]$. How can I solve this problem?


Definition (Riemann-sum theorem) Let $f(x)$ be a function dened on a closed interval $[a, b]$. Then, we have 
$$\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f\Big(a +(\frac{b - a}{n})i\Big)\frac{1}{n}=\int_{a}^{b}f(x)dx$$ 
 [/INST] \begin{align}\label{eq:7777}
 & \frac{2}{\sqrt{n-i} + \sqrt{n-i+1}} \leq \frac{1}{\sqrt{n-i}} \leq \frac{2}{\sqrt{n-i} + \sqrt{n-i-1}} \nonumber\\
 & \qquad \Rightarrow 2(\sqrt{n-i+1} - \sqrt{n-i}) \leq \frac{1}{\sqrt{n-i}} \leq 2(\sqrt{n-i} - \sqrt{n-i-1}) \nonumber\\
 & \qquad \Rightarrow 2 \sum_{i=1}^{n-1}(\sqrt{n-i+1} - \sqrt{n-i}) \leq \sum_{i=1}^{n-1} \frac{1}{\sqrt{n-i}} \leq 2 \sum_{i=1}^{n-1}(\sqrt{n-i} - \sqrt{n-i-1}) \nonumber\\
 & \qquad \Rightarrow 2 (\sqrt{n}-1)  \leq \sum_{i=1}^{n-1} \frac{1}{\sqrt{n-i}} \leq 2 \sqrt{n-1} 
 \end{align}
</s>