Evaluate the following limit: _ n 1^2+2^2+ +n^2 n^4 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{n}\rightarrow\infty}\frac{1^2+2^2+\ \dots+\text{n}^2}{\text{n}^4}$

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Answer

$\lim\limits_{\text{n}\rightarrow\infty}\frac{1^2+2^2+\ \dots+\text{n}^2}{\text{n}^4}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\Big[\frac12\text{n}(\text{n}+1)\Big]^2}{\text{n}^4}$ $\bigg[1^3+2^3+3^3+\ \cdots+\text{n}^3=\Big(\frac12\text{n}(\text{n}+1)\Big)^2\bigg]$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac{\frac14\text{n}^2(\text{n}+1)^2}{\text{n}^4}$
$=\lim\limits_{\text{n}\rightarrow\infty}\frac14\frac{\big(\text{n}^2\big(\text{n}^2+1+2\text{n}\big)\big)}{\text{n}^4}$ $\Big[\text{Multiplying the term }\frac{\infty}{\infty}\text{ from}\Big]$
$=\frac{1}{4}\lim\limits_{\text{n}\rightarrow{\infty}}\frac{\Big(1+\frac{1}{\text{n}^2}+\frac{2}{\text{n}}\Big)}{1}$
$=\frac{1}{4}$

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