$\lim_{n \rightarrow \infty}\left\lbrack \ {\frac{1}{n^{2}}^{\ }\sec^{2}{\frac{1}{n^{2}} + \frac{2}{n^{2}}\sec^{2}\frac{4}{n^{2}}}} + ... + \frac{n}{n^{2}}\sec^{2}1 \right\rbrack$equals

Question

Dr. Abubacker · Accepted Answer

(a)
Let $A = \lim_{n ightarrow \infty}\frac{1}{n}\left\lbrack \frac{1}{n}\sec^{2}{\left( \frac{1}{n} ight)^{2} + \frac{2}{n}\sec^{2}\left( \frac{2}{n} ight)^{2}} ight.\$
$+ ... + \frac{n}{n}\left. \ \sec^{2}\left( \frac{n}{n} ight)^{2} ightbrack$
$= \lim_{n ightarrow \infty}\frac{1}{n}\sum_{r = 1}^{n}{{\left( \frac{r}{n} ight)\sec^{2}}\left( \frac{r}{n} ight)^{2}}$
$\Rightarrow \ A = \int_{0}^{1}{x\sec^{2}{(x)^{2}dx}}$
Put $x^{2} = t \Longrightarrow x\ dx = \frac{dt}{2}$
$	herefore A = \frac{1}{2}\int_{0}^{1}{\sec^{2}{t\ dt = \frac{1}{2}\lbrack	an{t\left. \ \  ightbrack_{0}^{1}} = 	an 1}}$

Q. $\lim_{n \rightarrow \infty}\left\lbrack \ {\frac{1}{n^{2}}^{\ }\sec^{2}{\frac{1}{n^{2}} + \frac{2}{n^{2}}\sec^{2}\frac{4}{n^{2}}}} + ... + \frac{n}{n^{2}}\sec^{2}1 \right\rbrack$equals

Explanation

Reviewed by Dr. Abubacker

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