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 ...

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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

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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.

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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

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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