Evaluate _ n 1+2+3+ .+n n^2 . - Vidyadip

Question

Evaluate $\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots .+n}{n^2}$.

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Answer

$\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots .+n}{n^2}$
$\begin{array}{l}=\lim _{n \rightarrow \infty} \frac{\frac{n(n+1)}{2}}{n^2} \\ =\frac{1}{2} \lim _{n \rightarrow \infty} \frac{n^2+n}{n^2} \\ =\frac{1}{2} \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right) \\ =\frac{1}{2}(1+0) \\ =\frac{1}{2} \times 1=\frac{1}{2} .\end{array}$

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