So i have this limit:
$$\lim_{n\rightarrow \infty}\left(\frac{1}{n-n\sin^2(\frac{\pi}{4n})}+\frac{1}{n-n\sin^2(\frac{2\pi}{4n})}+ \cdots +\frac{1}{n-n\sin^2(\frac{n\pi}{4n})}\right)$$
So i translated it into this:
$$\lim_{n\rightarrow \infty} \left(\frac{1}{n} \sum_{k=0}^{n}\frac{1}{1-\sin^2(\frac{k\pi}{4n})}\right)$$
So i have an idea to use Riemann definition of integral to somehow translate it to integrals.
But i don't know how to script it.
Any help would be appreciated.
So in fact you got
$$\lim_{n\rightarrow \infty} \left(\frac{1}{n} \sum_{k=0}^{n}\frac{1}{1-\sin^2\frac{k\pi}{4n}}\right)=\lim_{n\rightarrow \infty} \left(\frac{1}{n} \sum_{k=0}^{n}\frac{1}{\cos^2\frac{k\pi}{4n}}\right)=\int_0^{\pi/4}\frac{dx}{\cos^2x}=1$$
Like The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$,
$$\lim_{n\rightarrow \infty} \left(\frac{1}{n} \sum_{k=0}^{n}\frac{1}{1-\sin^2(\frac{k\pi}{4n})}\right)=\int_0^1\dfrac{dx}{1-\sin^2\dfrac{x\pi}4}=\int_0^1\sec^2\dfrac{\pi x}4dx=?$$
