Question

Evaluate \[ \lim_{n\to\infty}\frac{1}{n} \left[ \tan^2\frac{\pi}{4n} +\tan^2\frac{2\pi}{4n} +\cdots+ \tan^2\frac{n\pi}{4n} \right]. \]

• A. \(1\)
• B. \(0\)
• C. \(-1\)
• D. \(\dfrac{4-\pi}{\pi}\)

Hint:

Whenever a limit contains \[ \frac1n\sum_{r=1}^{n}f\!\left(\frac{r}{n}\right), \] recognize it as a Riemann sum and convert it directly into \[ \int_0^1 f(x)\,dx. \]

Answer 1

The Correct Option is D