The general term for summation of n terms is given and the nth term is given .Finding the relation between them when n tends to infinity.
Let $$ S_n = \sum_{k=1}^n \frac{\tan^{-1} \frac{k}{n}}{n} \quad \text{and} \quad T_n = \sum_{k=0}^{n-1} \frac{\tan^{-1} \frac{k}{n}}{n} \quad \text{for}~n \in \mathbb{N} $$ then which of the following statements is false?
- (A) $S_n > \frac{\pi - \ln 4}{4}$
- (B) $T_n < \frac{\pi - \ln 4}{4}$
- (C) $\lim_{n \to \infty} S_n > \lim_{n \to \infty} T_n$
- (D) none of these
Note first that$$I:=\int_0^1\arctan xdx=[x\arctan x-\tfrac12\ln(1+x^2)]_0^1=\frac{\pi-\ln 4}{4}.$$As you noted, $S_n>I>T_n$ and $\lim_{n\to\infty}S_n=\lim_{n\to\infty}T_n=I$, so the answer is (C).
I doubt we can verify (A), (B) are true without evaluating $I$, but since $S_n-T_n=\frac{\pi}{4n}$, (C) is false. So if we exploit the fact that the question's options tell us at most one of (A)-(C) are false, we can avoid evaluating $I$, or even noticing an integral is comparable to the sequences $S_n,\,T_n$.