`(lim)_(n->oo){1/(2n+1)+1/(2n+2)++1/(2n+n)}` `I n\ (1/3)` b. `I n\ (2/3)` c. `I n\ (3/2)` d. `I n\ (4/3)`

Explanation

`ln (frac{3}{2}) `

`lim _{n -> infty}{frac{1}{2 n+1}+frac{1}{2 n+2}+ldots ldots ldots+frac{1}{2 n+n}} `

`=lim _{n -> infty} sum_{r=1}^{n} frac{1}{2 n+r} `

`=lim _{n -> infty} frac{1}{n} sum_{r=1}^{n} frac{1}{2+frac{r}{n}} `

let` frac{r}{n}=x `

`=int_{0}^{infty} frac{1}{2+x} d x `

`=[log (2+x)]_{0}^{infty} `

`=log 3-log 2 `

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