Evaluate the following limits: _ n n^2 1+2+3+ +n - Vidyadip

Question

Evaluate the following limits:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{1+2+3+\ \dots+\text{n}}$

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Answer

$\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{1+2+3+\ \dots+\text{n}}$$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{\frac{1}{2}\text{n}(\text{n}+1)}$ $\Big[\because1+2+3+\ \dots+\text{n}=\frac{\text{n}(\text{n}+1)}{2}\Big]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{\frac{1}{2}\text{n}(\text{n}+1)}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{2\text{n}^2}{\text{n}^2+\text{n}}$
$=2\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{\text{n}^2+\text{n}}$
$=2\lim\limits_{\text{n}\rightarrow\infty}\frac{\text{n}^2}{\text{n}^2\big(1+\frac{1}{\text{n}}\big)}$
$=2\times\frac{1}{1\times0}$
$=2$

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