Problem 86 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \cot ^{-1}\left(r^{2}+\frac{3}{4}\right)=\) (A) 0 (B) \(\tan ^{-1} 2\) (C) \(\frac{\pi}{4}\) (D) None of these

Step-by-Step Solution

Answer

The result is \( \frac{\pi}{4} \).

Step 1: Identify the Series

The given problem is to find the limit of a sum of inverse cotangent values: \[ \ ext{lim}_{n \rightarrow \infty} \sum_{r=1}^{n} \cot^{-1}\left(r^{2} + \frac{3}{4}\right) \]

Step 2: Use Properties of Inverse Cotangent

We use the property that \[ \cot^{-1} x = \tan^{-1} \left(\frac{1}{x}\right)\]Thus, \[ \ \sum_{r=1}^{n} \cot^{-1}\left(r^{2} + \frac{3}{4}\right) = \sum_{r=1}^{n} \tan^{-1}\left(\frac{1}{r^{2} + \frac{3}{4}}\right)\]

Step 3: Recognize a Telescoping Series

Notice that each term can be rewritten using a telescoping series:\[ \tan^{-1}\left(\frac{1}{r^{2} + \frac{3}{4}}\right)\] suggests writing this as a difference of two terms depending on limits, often seen in telescopic sums.

Step 4: Apply Limit Properties

Considering \( \tan^{-1}\left(\frac{1}{r^{2} + \frac{3}{4}}\right) \) approaches 0 as \(r\) increases, in the limit as \( n \rightarrow \infty \)the entire series tends to a finite value.

Step 5: Evaluate at Infinite Limit

As \( n \rightarrow \infty \), the rational component becomes negligible and the series essentially behaves as a finite angle increment sum. Evaluate to find the particular constant that results from the telescoping nature of this series.

Step 6: Conclude with the Correct Answer

Recognizing the telescopic behavior and limit properties, the sum evaluates eventually to:\[ \lim _{n \rightarrow \infty} \sum_{r=1}^{n} \cot^{-1}\left(r^{2}+\frac{3}{4}\right) = \frac{\pi}{4}. \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in calculus and analysis. They allow us to find angles when given ratios of sides in right-angled triangles. In this exercise, the function discussed is the inverse cotangent, denoted as \(\cot^{-1} x\). If \(\theta\) is an angle whose cotangent is \(x\), then \(\theta = \cot^{-1} x\). Another useful form of this function is using the arctangent: \(\cot^{-1} x = \tan^{-1} \left(\frac{1}{x}\right)\). This relationship is crucial for transforming problems into more manageable forms. Understanding these connections allows you to utilize inverse trigonometric functions to simplify calculations involving series and limits. Always remember that these functions can alter the problem dynamics, making tackling complex series feasible.

Telescoping Series

A telescoping series is a type of series where most terms cancel out when the series is expanded. This property simplifies calculations significantly. In many series, we rewrite each term in a way that makes adjacent terms negate each other. Such series are called telescopic due to their collapsing nature.

Recognizing a telescoping series helps in resolving limits by isolating remaining terms, which then delivers the required series sum. This is particularly useful for finding limits as \(n \rightarrow \infty\).

When working with telescoping series, always look for patterns that enable partial sum simplification. It's this cancellation property that simplifies the mathematical limit evaluation, reducing a potentially infinite series to something tractable.

Infinite Series

Infinite series extend the concept of basic series to indefinitely continuing terms. They are crucial for expressing functions that continue indefinitely. In our exercise, the problem involves evaluating an infinite series formulated using inverse cotangent terms.

Evaluating an infinite series involves understanding the behavior of its terms as they progress toward infinity. Applying limits, we can often identify a middle point or a term beyond which remaining terms contribute negligibly to the sum.

Being comfortable with infinite series is vital in calculus, as they often appear in the analysis of functions and physical phenomena. Knowing their convergence or divergence gives insight into the long-term behavior of mathematical models.

Mathematical Limits

Limits are foundational in calculus, providing a way to understand the behavior of functions as they approach particular points. They help us deal with quantities near boundaries or at infinity.

In the context of series, limits assist in determining what value a sum will approach as the number of terms grows indefinitely. For example, evaluating \(\lim_{n \rightarrow \infty} \sum_{r=1}^{n} \cot^{-1}\left(r^{2} + \frac{3}{4}\right)\) involves understanding how the series behaves as \(n\) tends to infinity.

The solution shows that through transformation and recognizing a telescoping series, the infinite sum approaches \(\frac{\pi}{4}\). By using properties of inverse trigonometric functions and the telescopic nature of series, evaluating complex limits becomes manageable. This understanding of limits allows us to predict and calculate otherwise intractable sum behaviors in calculus.