The value of lim n ( lim x 0 - r = 1 2 n + 1 [ x r ] ) + n + 3 n ( 1 + n ) n - n ( n ) n , where [ · ] denotes greatest integer function

Question

The value of $\lim_{n \to \infty} \frac{(\lim_{x \to 0^{-}} \sum_{r = 1}^{2 n + 1} [x^{r}]) + n + 3}{ℓ n {(1 + n)}^{n} - ℓ n {(n)}^{n}}$ , where $[\cdot]$ denotes greatest integer function

Smart Achievers · Accepted Answer

(2)

$\lim_{x \to 0^{-}} \sum_{r = 1}^{2 n + 1} [x^{r}] = \lim_{x \to 0^{-}} [x] + [x^{2}] + [x^{3}] + \dots + [x^{2 n + 1}]$

$= (- 1) + 0 + (- 1) + \dots + 0 + (- 1) = (- 1) (n + 1) = - n - 1$

$\Rightarrow L = \lim_{n \to \infty} \frac{- n - 1 + n + 3}{n \ln (\frac{1 + n}{n})}$

$= \lim_{n \to \infty} \frac{2 (\frac{1}{n})}{\ln (1 + \frac{1}{n})}$ $\Rightarrow \frac{1}{n} = t$

$\Rightarrow L = \lim_{t \to 0} \frac{2 t}{\ln (1 + t)} = 2$

Solution

Q.
The value of $\lim_{n \to \infty} \frac{(\lim_{x \to 0^{-}} \sum_{r = 1}^{2 n + 1} [x^{r}]) + n + 3}{ℓ n {(1 + n)}^{n} - ℓ n {(n)}^{n}}$ , where $[\cdot]$ denotes greatest integer function

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Solution

Q. The value of limn→∞(limx→0-∑r=12n+1[xr])+n+3ℓn(1+n)n-ℓn(n)n, where [·] denotes greatest integer function

Ans. (2) limx→0-∑r=12n+1[xr]=limx→0-[x]+[x2]+[x3]+⋯+[x2n+1] =(-1)+0+(-1)+⋯+0+(-1)=(-1)(n+1)=-n-1 ⇒L=limn→∞-n-1+n+3nln(1+nn) =limn→∞2(1n)ln(1+1n)⇒1n=t ⇒L=limt→02tln(1+t)=2

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Q.
The value of $\lim_{n \to \infty} \frac{(\lim_{x \to 0^{-}} \sum_{r = 1}^{2 n + 1} [x^{r}]) + n + 3}{ℓ n {(1 + n)}^{n} - ℓ n {(n)}^{n}}$ , where $[\cdot]$ denotes greatest integer function