Animated Solution for Mathematics - Definite Integration: ${\lim }_{n\to \infty }\left[\frac{1}{n^2}{\sec }^2\frac{1}{n^2}+\frac{2}{n^2}{\sec }^2\frac{4}{n^2}+\cdots +\frac{1}{n}{\sec }^{2}1\right]$ equals

The problem presents a series involving terms like $\frac{1}{n^2}{\sec }^2\frac{1}{n^2}$ and $\frac{2}{n^2}{\sec }^2\frac{4}{n^2}$. When dealing with a limit of a sum as $n\to \infty$, the Riemann Sum approach is the most powerful tool in our arsenal.

We identify the general term $T_r$ by observing the pattern in the numerators ($1,2,\dots ,n$) and the arguments of the secant function ($\frac{1}{n^2},\frac{4}{n^2},\dots ,\frac{n^2}{n^2}$). The $r$-th term is given by:

To apply the Riemann sum definition, we must express the series in the form ${\lim }_{n\to \infty }\frac{1}{n}{\sum }_{r=1}^{n}f\left(\frac{r}{n}\right)$. We manipulate the expression by factoring out $\frac{1}{n}$:

This reveals our function to be $f(x)=x{\sec }^2(x^2)$. The structure is now perfectly aligned for the transition from discrete summation to continuous integration.

We replace $\frac{r}{n}$ with $x$ and $\frac{1}{n}$ with $dx$. As $n\to \infty$, the lower limit $r=1$ approaches $0$, and the upper limit $r=n$ approaches $1$.

The limit of the sum transforms into the following definite integral:

To solve this integral, we employ the method of substitution. Let $u=x^2$, which implies $du=2xdx$, or $xdx=\frac{1}{2}du$.

Adjusting the limits of integration: when $x=0$, $u=0$; when $x=1$, $u=1$. The integral becomes:

Evaluating the integral, we know that the antiderivative of ${\sec }^2(u)$ is $\tan (u)$. Applying the fundamental theorem of calculus:

Since $\tan (0)=0$, the final result is: