how can I write this sum in sigma notation?

Question

I find it difficult to write this in sigma notation. I tried but couldn't figure out.

$$ \frac{1}{n} \sqrt{1-\left(\frac{0}{n}\right)^2} + \frac{1}{n} \sqrt{1-\left(\frac{1}{n}\right)^2} + \dots + \frac{1}{n} \sqrt{1-\left(\frac{n-1}{n}\right)^2} $$

Answer 1

It is more or less finding the pattern as to what changes and what does not:

\begin{align*} &\frac{1}{n} \sqrt{1-\left(\frac{0}{n}\right)^2} + \frac{1}{n} \sqrt{1-\left(\frac{1}{n}\right)^2} + \dots + \frac{1}{n} \sqrt{1-\left(\frac{n-1}{n}\right)^2} \\ &= \frac{1}{n} \sqrt{1-\left(\frac{\color{blue}{\boxed{0}}}{n}\right)^2} + \frac{1}{n} \sqrt{1-\left(\frac{\color{blue}{\boxed{1}}}{n}\right)^2} + \dots + \frac{1}{n} \sqrt{1-\left(\frac{\color{blue}{\boxed{n-1}}}{n}\right)^2} \\ &= \Biggl[ \text{Sum of } \frac{1}{n} \sqrt{1-\left(\frac{\color{blue}{\blacksquare}}{n}\right)^2} \text{'s, where $\color{blue}{\blacksquare}$ runs over $0, 1, \dots, n-1$} \Biggr] \\ &= \sum_{\color{blue}{\blacksquare}=0}^{n-1} \frac{1}{n} \sqrt{1-\left(\frac{\color{blue}{\blacksquare}}{n}\right)^2} \end{align*}

Now replace the placeholder $\color{blue}{\blacksquare}$ by another variable, say $k$ for instance, if you want to make it look fancier.