Limn-n-n2-1-n-n2-2-n-n2-3-n-n2-n is

The correct option is A 0We have, n→∞lim nn^2+1+nn^2+2+nn^2+3+—–+nn^2+n #10; =n→∞lim nn^2+n #10; =n→∞lim nn[ #10;1n+1n] =n→∞lim [ 1n+1 ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Differentiation under Integral Sign

limn→∞n/n2+1+...

Question

${lim}_{n \to \infty} \frac{n}{n^{2} + 1} + \frac{n}{n^{2} + 2} + \frac{n}{n^{2} + 3} + . . . . + \frac{n}{n^{2} + n}$ is

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\infty

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is A 0
We have,
${lim}_{n \to \infty} \frac{n}{n^{2} + 1} + \frac{n}{n^{2} + 2} + \frac{n}{n^{2} + 3} + - - - - - + \frac{n}{n^{2} + n}$
$= {lim}_{n \to \infty} \frac{n}{n^{2} + n}$
$= {lim}_{n \to \infty} \frac{n}{n} ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{n + \frac{1}{n}} ⎤ ⎥ ⎥ ⎥ ⎦$
$= {lim}_{n \to \infty} ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{n + \frac{1}{n}} ⎤ ⎥ ⎥ ⎥ ⎦$
$= {lim}_{n \to \infty} ⎡ ⎢ ⎢ ⎢ ⎣ \frac{1}{\infty + \frac{1}{\infty}} ⎤ ⎥ ⎥ ⎥ ⎦$
$= \frac{1}{\infty}$
$= 0$

Hence, this is the answer.

Suggest Corrections

Similar questions

φ (1) ⌊ \frac{n}{1} ⌋ + φ (2) ⌊ \frac{n}{2} ⌋ + . . . + φ (n) ⌊ \frac{n}{n} ⌋ = \frac{n (n + 1)}{n},

where

φ

denotes Euler's totient function.

Let

p

be a prime number &

n

be a positive integer, then exponent of prime

p

n!

is denoted by

E_{p} (n!)

& is given by

E_{p} (n!) = [\frac{n}{p}] + [\frac{n}{p^{2}}] + [\frac{n}{p^{3}}] + . . . . . + [\frac{n}{p^{x}}]

where

x

is the largest positive integer such that

p^{x} \leq n < p^{x + 1}

and

[\cdot]

denotes the greatest integer

Again every natural number

N

can be expressed as the product of its prime factors given by

N = P_{1}^{k_{2}} P_{2}^{k_{2}} . . . . P_{r}^{k_{r}}

where

P_{1}, P_{2}, P_{3}, . . . . . . . P_{r}

are prime numbers &

k_{1}

are whole numbers.

The greatest integer

n

for which

77!

is divisible by

3^{n}

Q. If

3^{n}

is a factor of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 & 2 & 2 n_{C_{1}} & n + 3_{C_{1}} & n + 6_{C_{1}} \frac{n_{C_{2}}}{2} & \frac{n + 3_{C_{2}}}{2} & \frac{n + 6_{C_{2}}}{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

, then the maximum value of ‘n’ is:

Q. If

n

is even positive integer, then the condition that the greatest term in the expansion of

(1 + x)^{n}

may also have the greatest coefficient is

Q. Let

n ϵ N

& the A.M., G.M., H.M. & the root mean square of

n

numbers

2 n + 1, 2 n + 2, . . .,

up to

n^{t h}

number are

A_{n}

G_{n}

H_{n}

and

R_{n}

respectively.
On the basis of above information answer the following questions

lim n \to \infty \frac{n}{H_{n}}

equals

Join BYJU'S Learning Program

limn→∞nn2+1+nn2+2+nn2+3+....+nn2+n is

The correct option is A 0We have,limn→∞nn2+1+nn2+2+nn2+3+−−−−−+nn2+n=limn→∞nn2+n=limn→∞nn⎡⎢ ⎢ ⎢⎣1n+1n⎤⎥ ⎥ ⎥⎦=limn→∞⎡⎢ ⎢ ⎢⎣1n+1n⎤⎥ ⎥ ⎥⎦=limn→∞⎡⎢ ⎢ ⎢⎣1∞+1∞⎤⎥ ⎥ ⎥⎦=1∞=0 Hence, this is the answer.

${lim}_{n \to \infty} \frac{n}{n^{2} + 1} + \frac{n}{n^{2} + 2} + \frac{n}{n^{2} + 3} + . . . . + \frac{n}{n^{2} + n}$ is