Interesting Limit (22)

Given $f_n(x)=\sum_{i=0}^n\frac{1}{i}\sin(ix)$ , we can look at its derivative to find candidates for the maximum. Its derivative is written as $\sum_{i=1}^n\cos(ix)$ , the first root of which is always at $x=\frac{\pi}{n+1}$ for $n>0$ . Each function $f_n$ looks like a descending slope, with initially large oscillations that dampen for higher values of x, so the maximum is always at the first root of the derivative. We plug that value into $f_n$ to get the sum $\sum_{i=1}^n\frac{\sin(\pi\frac{i}{n+1})}{i}$ . Given that for large n, each successive term is rather small, we can extend the summation region by 1, set $m=n+1$ , and multiply by $\frac{\frac{1}{m}}{\frac{1}{m}}$ to get the new sum $\sum_{i=1}^m\frac{1}{m}\frac{\sin(\pi \frac{i}{m})}{\frac{i}{m}}$ . This is the formula for a Riemann sum from 0 to 1 of $\frac{\sin(\pi x)}{x}$ , and doing a little scaling we get $\int_0^\pi \frac{\sin(x)}{x}dx$ .