Problem 197 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{n \rightarrow \infty}\left[\frac{1}{n^{2}} \sec ^{2} \frac{1}{n^{2}}+\frac{2}{n^{2}} \sec ^{2} \frac{4}{n^{2}}+\ldots .+\frac{1}{n^{2}} \sec ^{2} 1\right]\) equals [2005] (A) \(\frac{1}{2} \sec 1\) (B) \(\frac{1}{2} \operatorname{cosec} 1\) (C) \(\tan 1\) (D) \(\frac{1}{2} \tan 1\)

Step-by-Step Solution

Answer

The limit equals (C) \(\tan 1\).

Step 1: Write the Limit as a Sum

The expression inside the limit is a form of a Riemann sum. It can be rewritten as \[\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{n^{2}} \sec^{2} \left( \frac{k^{2}}{n^{2}} \right)\]where each term is \(\frac{1}{n^{2}} \sec^{2} \left( \frac{k^{2}}{n^{2}} \right)\), characterized by discrete steps.

Step 2: Identify the Function and Step

Identify \(f(x) = \sec^{2}(x^{2})\). The term \(\frac{1}{n^{2}}\) acts like \(\Delta x\) and the expression as \(n\) approaches infinity represents \(\int_{0}^{1} f(x) \, dx\).Here, \(\Delta x = \frac{1}{n}\) not \(\frac{1}{n^{2}}\). Thus, rewrite the terms in a way that the step size \(\frac{1}{n}\) properly represents the partitioning needed for Riemann summation.

Step 3: Proper Riemann Representation

Given the interval \([0,1]\), rewrite:\[\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{n} \cdot \sec^{2} \left(\frac{k}{n}\right)\] where \(\frac{1}{n}\) is the width of each partitioned rectangle and \(\sec^{2}\left(\frac{k}{n}\right)\) is the height. This shows we're taking an integral of \(\sec^{2}(x)\) over \([0, 1]\).

Step 4: Evaluate the Integral

The rewriting suggests the integral:\[\int_{0}^{1} \sec^{2}(x) \, dx\]The antiderivative of \(\sec^{2}(x)\) is \(\tan(x)\). Evaluate it from 0 to 1:\[\left[ \tan(x) \right]_{0}^{1} = \tan(1) - \tan(0)\]Since \(\tan(0) = 0\), the result simplifies to \(\tan(1)\).

Step 5: Select the Correct Answer

The integral evaluates to \(\tan(1)\), which corresponds to option (C) \(\tan(1)\).

Key Concepts

Riemann Sum

In mathematics, a Riemann sum is an approach for approximating the total area under a curve, represented by the integral of a function. It is essential in the field of calculus to understand the concept of integration. The idea is simple: divide the region into small rectangles and then sum up their areas.
Imagine having a curve and trying to calculate the area beneath it. The Riemann sum method breaks the time domain into smaller subintervals. In this problem, each term of the sum takes the form \(\frac{1}{n^{2}} \sec^{2} \left(\frac{k^{2}}{n^{2}}\right)\). By increasing the number of rectangles (or steps) \(n\), the approximation becomes more precise. As \(n\) goes to infinity, the Riemann sum becomes a definite integral.

• Each rectangle has a width \(\Delta x = \frac{1}{n}\).
• The height for each rectangle at a particular \(k\) is determined by the function value \(\sec^{2}(x)\).

This process will ultimately lead us to the concept of definite integrals, translating Riemann sums into precise area measurements.

Definite Integrals

Definite integrals form the foundation for calculating the area under a curve precisely. When we evaluate the definite integral of a function over an interval, we're essentially summing up infinitely many infinitesimally small areas.
In this exercise, once we recognized the expression as a Riemann sum, the next step was to convert it into a definite integral. The problem transitions from summing rectangles to evaluating the integral \(\int_{0}^{1} \sec^{2}(x) \, dx\).

• This integral describes the area under the curve of \(\sec^{2}(x)\) from \(x = 0\) to \(x = 1\).
• Integrating \(\sec^{2}(x)\) requires finding an antiderivative, which is \(\tan(x)\).

Finally, by using the limits of integration, evaluate the integral from 0 to 1, leading us to \(\tan(1) - \tan(0) = \tan(1)\). This shows how the Riemann sum facilitated a transition from approximation to exact calculation using definite integrals.

Trigonometric Functions

Trigonometric functions are fundamental in calculus, often appearing in problems involving periodic functions or wave-like behavior. They enable transformations and simplifications in integrals and derivatives due to their unique properties.
The trigonometric function involved in this exercise was \(\sec^2(x)\), directly related to the tangent function. The secant function is the reciprocal of the cosine function. Therefore, \(\sec^2(x)\) is \(\left(\frac{1}{\cos(x)}\right)^2\). Here's how these functions play a role:

• The derivative of \(\tan(x)\) is \(\sec^{2}(x)\).
• Recognizing the integral of \(\sec^{2}(x)\) helped identify that its antiderivative is \(\tan(x)\).

Using these properties allows one to switch between differentiated and integrated forms seamlessly. This exercise's solution leveraged this relationship to simplify the path towards determining the definite integral value.