The value of lim _{n rightarrow infty} frac{1 | vedclass

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(B) The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.Substituting this into the limit expression

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$\frac{1}{2}$

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(B) The sum of the first $n$ natural numbers is given by the formula $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$.Substituting this into the limit expression:$\lim _{n \rightarrow \infty} \frac{n(n+1)}{2n^{2}}$$= \lim _{n \rightarrow \infty} \frac{n^2+n}{2n^2}$$= \lim _{n \rightarrow \infty} \frac{1 + \frac{1}{n}}{2}$As $n \rightarrow \infty$,$\frac{1}{n} \rightarrow 0$.Therefore,the limit is $\frac{1+0}{2} = \frac{1}{2}$.

The value of $\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{n^{2}}$ is

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