The value of limn -r-n2n-12 1-r is-

The correct option is A 2 f(n)=1/√(n^2)+1/√(n^2+1)+1/√(n^2+2)+....+1/√(n^2+2n+1) Total number terms = 2n + 2 =2n+2/√(n^2+2n+1)≤ f(n) ≤2n+2/√(n^2) lim_n →∞2n+2/ ...

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

Byju's Answer

Standard XII

Mathematics

Definite Integral as Limit of Sum

The value of ...

Question

The value of ${lim}_{n \to \infty} \sum_{r = n^{2}}^{(n + 1)^{2}} (\frac{1}{\sqrt{r}})$ is___.

2

Open in App

Solution

The correct option is A
2

$f (n) = \frac{1}{\sqrt{n^{2}}} + \frac{1}{\sqrt{n^{2} + 1}} + \frac{1}{\sqrt{n^{2} + 2}} + . . . . + \frac{1}{\sqrt{n^{2} + 2 n + 1}}$
Total number terms = 2n + 2
$= \frac{2 n + 2}{\sqrt{n^{2} + 2 n + 1}} \leq f (n) \leq \frac{2 n + 2}{\sqrt{n^{2}}} {lim}_{n \to \infty} \frac{2 n + 2}{\sqrt{n^{2} + 2 n + 1}} = 2 {lim}_{n \to \infty} \frac{2 n + 2}{\sqrt{n^{2}}} = 2 S O {lim}_{n \to \infty} \sum_{r = n^{2}}^{(n + 1)^{2}} (\frac{1}{\sqrt{r}}) = 2$

Suggest Corrections

Similar questions

Q. What is the value of

{lim}_{n \to \infty} {(1 - \frac{1}{n})}^{2 n}

Q. The value of

lim n \to \infty n \prod r = 1 {(\frac{n^{2} + r^{2}}{n^{2}})}^{\frac{1}{n}}

is:

{lim}_{n \to \infty} (\frac{1}{n} + \frac{n^{2}}{(n + 1)^{3}} + \frac{n^{2}}{(n + 2)^{3}} + \dots + \frac{1}{8 n})

is equal to

Q. The value of

lim n \to \infty \frac{n \sum r = 1 r + 2. n - 1 \sum r = 1 r + 3. n - 2 \sum r = 1 r + . . . . . . + n .1}{n^{4}}

Q. The value of

lim n \to \infty n \sum r = 1 \frac{1}{n} \sqrt{\frac{n + r}{n - r}}

Join BYJU'S Learning Program

The value of limn→∞∑(n+1)2r=n2(1√r) is___. 2

The correct option is A 2 f(n)=1√n2+1√n2+1+1√n2+2+....+1√n2+2n+1 Total number terms = 2n + 2 =2n+2√n2+2n+1≤f(n)≤2n+2√n2limn→∞2n+2√n2+2n+1=2limn→∞2n+2√n2=2SOlimn→∞∑(n+1)2r=n2(1√r)=2

The value of ${lim}_{n \to \infty} \sum_{r = n^{2}}^{(n + 1)^{2}} (\frac{1}{\sqrt{r}})$ is___.

2