mathop {lim }limits_{n to infty } left[ {frac | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The given expression is $\mathop {\lim }\limits_{n 	o \infty } \left[ {\frac{1 + 2 + 3 + \dots + n}{1 - {n^2}}} ight]$.Using the formula for th

Vedclass · Accepted Answer

$-\frac{1}{2}$

Vedclass · Answer

(B) The given expression is $\mathop {\lim }\limits_{n 	o \infty } \left[ {\frac{1 + 2 + 3 + \dots + n}{1 - {n^2}}} ight]$.Using the formula for the sum of the first $n$ natural numbers,$\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$,the expression becomes:$\mathop {\lim }\limits_{n 	o \infty } \frac{n(n+1)}{2(1 - {n^2})}$$= \mathop {\lim }\limits_{n 	o \infty } \frac{n^2 + n}{2 - 2n^2}$Dividing the numerator and denominator by $n^2$:$= \mathop {\lim }\limits_{n 	o \infty } \frac{1 + \frac{1}{n}}{\frac{2}{n^2} - 2} = \frac{1 + 0}{0 - 2} = -\frac{1}{2}$.

$\mathop {\lim }\limits_{n \to \infty } \left[ {\frac{1}{{1 - {n^2}}} + \frac{2}{{1 - {n^2}}} + \frac{3}{{1 - {n^2}}} + \dots + \frac{n}{{1 - {n^2}}}} \right] =$

Similar Questions

If $[x]$ denotes the greatest integer $\leq x$,then $\lim_{n \rightarrow \infty} \frac{1}{n^3} \{[1^2 x] + [2^2 x] + [3^2 x] + \ldots + [n^2 x] \} = $

$\lim _{x}$ ${\rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots \cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}} = \dots$

$\mathop {\lim }\limits_{x \to 2} \frac{|x - 2|}{x - 2} = $

For the function $f(x) = (1 + \frac{1}{x})^x$,which of the following limits tends to unity?

If $\lim _{x \rightarrow 0} \frac{\cos 2x - \cos 4x}{1 - \cos 2x} = k$,then $\lim _{x \rightarrow k} \frac{x^k - 27}{x^{k+1} - 81} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module