lim n → ∞{1/(2n + 1) + 1/(2n +2)+....1/(2n + n)} is equal to A. In(1/3) B. In (2/3) C. In (3/2) D. In(4/2)

Question

\(\lim\limits_{n \to \infty}\Big\{\cfrac{1}{2n+1}+\cfrac{1}{2n+2}+....+\cfrac{1}{2n+n}\Big\} \)is equal to

A. In \(\Big(\cfrac13\Big)\)

B. In \(\Big(\cfrac23\Big)\)

C. In \(\Big(\cfrac32\Big)\)

D. In  \(\Big(\cfrac42\Big)\)

Answer 1

Correct option is C. In\(\Big(\cfrac32\Big)\) 

Given, \(\lim\limits_{n \to \infty}\Big\{\cfrac{1}{2n+1}+\cfrac{1}{2n+2}+....+\cfrac{1}{2n+n}\Big\} \)

Now for easy solvation, replace \(\lim\limits_{n \to \infty} \sum_{r=1}^{n} \) with \(\int\limits_{0}^{1} \), \(\cfrac{r}n\) with x and \(\cfrac1n\) with dx