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

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(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

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$-\frac{1}{2}$

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(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 $n^2$:$= \mathop {\lim }\limits_{n 	o \infty } \frac{1 + \frac{1}{n}}{\frac{2}{n^2} - 2} = \frac{1 + 0}{0 - 2} = -\frac{1}{2}$.

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

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