MCQ
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+\ldots+\frac{n}{n^2}\right)$ is
- ✓$\frac{1}{2}$
- B
- C1
- D$-\frac{1}{2}$
✓
Answer
Correct option: A.
$\frac{1}{2}$
(A)
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+\ldots+\frac{n}{n^2}\right)$
$=\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+n}{n^2}\right)=\lim _{n \rightarrow \infty} \frac{\frac{n(n+1)}{2}}{n^2}$
$=\frac{1}{2} \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)=\frac{1}{2}$
Alternate method:
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+\ldots .+\frac{n}{n^2}\right)$
$=\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+n}{n^2}\right)$
$\lim _{x \rightarrow \infty} \frac{1^\alpha+2^\alpha+3^\alpha+\ldots+x^\alpha}{x^{\alpha+1}}=\frac{1}{\alpha+1}$
$=\frac{1}{1+1} $
$=\frac{1}{2}$
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+\ldots+\frac{n}{n^2}\right)$
$=\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+n}{n^2}\right)=\lim _{n \rightarrow \infty} \frac{\frac{n(n+1)}{2}}{n^2}$
$=\frac{1}{2} \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)=\frac{1}{2}$
Alternate method:
$\lim _{n \rightarrow \infty}\left(\frac{1}{n^2}+\frac{2}{n^2}+\frac{3}{n^2}+\ldots .+\frac{n}{n^2}\right)$
$=\lim _{n \rightarrow \infty}\left(\frac{1+2+3+\ldots+n}{n^2}\right)$
$\lim _{x \rightarrow \infty} \frac{1^\alpha+2^\alpha+3^\alpha+\ldots+x^\alpha}{x^{\alpha+1}}=\frac{1}{\alpha+1}$
$=\frac{1}{1+1} $
$=\frac{1}{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If $f:[0, \pi / 2) \rightarrow R$ is defined as
$f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$. Then the range of f is
→$f(\theta)=\left|\begin{array}{ccc}1 & \tan \theta & 1 \\ -\tan \theta & 1 & \tan \theta \\ -1 & -\tan \theta & 1\end{array}\right|$. Then the range of f is
The probability that a leap year will have 53 Fridays or 53 Saturdays, is
If $z$ is a complex number such that $\frac{z-1}{z+1}$ is purely imaginary, then
→The inequality representing the following graph is :
The sum of focal distances of any point on the ellipse with major and minor axes as 2 a and 2 b respectively, is equal to
For a frequency distribution standard deviation is computed by applying the formula : $Z$
→Two dice are thrown together. The probability that neither they show equal digits nor the sum of their digits is $9$ will be :
→The equation of the ellipse with focus $(-1, 1),$ directrix $x - y + 3 = 0$ and eccentricity $\frac{1}{2}$ is :
→The value of $\sin 1^{\circ}+\sin 2^{\circ}+\ldots+\sin 359^{\circ}$ is equal to
→If $\text{z}=1-\cos\theta+\text{i}\sin\theta,$ then $|\text{z}|=$
→