Let Pk be a point in x y plane whose x coordinate is 1+(k/n)(k=1,2,3, ldots ldots, n) on the curve y= ln x. If A is (1,0), then undersetn arrow ∞ textLim (1/n) displaystyle∑k=1n( AP k )2 equals

Question

Let Pk be a point in x y plane whose x coordinate is 1+(k/n)(k=1,2,3, ldots ldots, n) on the curve y= ln x. If A is (1,0), then undersetn arrow ∞ textLim (1/n) displaystyle∑k=1n( AP k )2 equals (A) (1/3)+2 ln 2 2 (B) (1/3)+2 ln 2((2/ e )) (C) (1/3)+ ln 2((2/ e )) (D) (1/3)+2 ln ((2/ e ))

Tardigrade Admin · Accepted Answer

L = undersetn arrow ∞ textLim (1/ n ) displaystyle∑ k =14((( k / n ))2+ ln 2(1+( k / n ))) =∫ limits01(x2+ ln 2(1+x)) d x=(1/3)+2 ln 2 (2/e)

Q. Let $P_{k}$ be a point in $x y$ plane whose $x$ coordinate is $1 + \frac{k}{n} (k = 1, 2, 3, \dots\dots, n)$ on the curve $y = ln x$ . If $A$ is $(1, 0)$ , then $n \to \infty Lim \frac{1}{n} k = 1 \sum n (A P_{k})^{2}$ equals

$\frac{1}{3} + 2 ln^{2} 2$

$\frac{1}{3} + 2 ln^{2} (\frac{2}{e})$

$\frac{1}{3} + ln^{2} (\frac{2}{e})$

$\frac{1}{3} + 2 ln (\frac{2}{e})$

$L = n \to \infty Lim \frac{1}{n} k = 1 \sum 4 ((\frac{k}{n})^{2} + ln^{2} (1 + \frac{k}{n}))$
$= 0 \int 1 (x^{2} + ln^{2} (1 + x)) d x = \frac{1}{3} + 2 ln^{2} \frac{2}{e}$

Q. Let Pk​ be a point in xy plane whose x coordinate is 1+nk​(k=1,2,3,……,n) on the curve y=lnx. If A is (1,0), then n→∞Lim​n1​k=1∑n​(APk​)2 equals

31​+2ln22

31​+2ln2(e2​)

31​+ln2(e2​)

31​+2ln(e2​)