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  4. displaystyle limn → ∞((1/n2)+(3/n2)+(5/n2)+.....+(2n+1/n2)) is equal to

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Q. $\displaystyle \lim_{n \to \infty}$$\left(\frac{1}{n^{2}}+\frac{3}{n^{2}}+\frac{5}{n^{2}}+.....+\frac{2n+1}{n^{2}}\right)$ is equal to

Limits and Derivatives

Solution:

$\displaystyle \lim_{n \to \infty}$$\left(\frac{1}{n^{2}}+\frac{3}{n^{2}}+\frac{5}{n^{2}}+.......+\frac{2n+1}{n^{2}}\right)$
$=\displaystyle \lim_{n \to \infty}$$\frac{1}{n^{2}}\left(1+3+5+.......+\left(2n+1\right)\right)$
$=\displaystyle \lim_{n \to \infty}$$\frac{\left(n+1\right)^{2}}{n^{2}}$
$=\displaystyle \lim_{n \to \infty}$$\frac{n^{2}+1+2n}{n^{2}}$
$=\displaystyle \lim_{n \to \infty}$$\left(1+\frac{1}{n^{2}}+\frac{2}{n}\right)=1$