Question

For each positive integer \(n\), let $$ y_{n}=\frac{1}{n}((n+1)(n+2) \cdots(n+n))^{\frac{1}{n}} $$ For \(x \in \mathbb{R}\), let \([x]\) be the greatest integer less than or equal to \(x .\) If \(\lim _{n \rightarrow \infty} y_{n}=L\), then the value of \([L]\) is

Short answer

[L] = 1

Step-by-step solution

Recognizing the limit definition

Observe that the expression for \(y_{n}\) involves computing the geometric mean of the first \(n\) consecutive integers starting from \(n+1\). We want to find the value of this mean as \(n\) approaches infinity (\(n\) becomes very large).

Simplifying the expression for \(y_{n}\)

Recall that the geometric mean of numbers is the \(n\)th root of the product of \(n\) numbers. To make the expression more manageable, factor out \(n^n\) from the numerator as all terms have \(n\) as a factor: $$y_{n} = \frac{1}{n}((n+1)(n+2) \cdots(n+n))^\frac{1}{n} = \frac{1}{n}(n^n(1 + \frac{1}{n})(1 + \frac{2}{n}) \cdots (1 + \frac{n}{n}))^\frac{1}{n}.$$

Applying limit to the modified expression for \(y_{n}\)

As \(n \rightarrow \infty\), each term inside the parentheses of the form \((1 + \frac{k}{n})\) for \(k=1,2,...,n\), approaches 1 because \(\frac{k}{n} \rightarrow 0\). Therefore, the product of such terms also approaches 1. Thus the limit of \(y_{n}\) as \(n \rightarrow \infty\) greatly simplifies.

Calculating the limit of \(y_{n}\) as \(n \rightarrow \infty\)

Simplifying further, we get: $$\lim_{n \to \infty}y_{n} = \lim_{n \to \infty}\frac{1}{n} \cdot n = \lim_{n \to \infty}1 = 1.$$ Therefore, the limit \(L = 1\).

Determining the greatest integer less than or equal to \(L\)

Since \(L=1\) and \(L\) is already an integer, the greatest integer less than or equal to \(L\) is 1 itself. Thus, \([L] = [1] = 1\).

Key concepts

Limit Definition

Understanding the concept of limits is crucial when dealing with functions involving variables that approach a particular value. The limit of a sequence or function describes what value the sequence or function approaches as the variable gets closer and closer to a certain number or infinity. In mathematics, particularly in calculus, this is a fundamental concept that allows analysis of behavior of functions without requiring the exact value at that point. The notation \( \lim_{n \to \infty} a_n = L \) signifies that as 'n' grows without bound, the value of the sequence \(a_n\) approaches 'L'.

In practical terms, the limit helps us handle situations involving growth to infinity or infinitesimal closeness, which are essential for understanding continuous functions and analyzing the rates at which quantities change. This forms the basis for differentiation and integration, the core of calculus.

Simplifying Expressions

Simplifying expressions is a process used to make a mathematical expression easier to understand and work with. It can involve factoring, combining like terms, and applying various algebraic rules. The goal is to reduce the complexity of the expression without changing its value. This practice is not only beneficial for hand calculations but is also imperative for solving problems efficiently, especially when dealing with limits and an increasing computational complexity as the variable grows larger.

During the simplification, recognizing common factors and using algebraic identities can lead to a form that reveals properties of the expression, such as its behavior at infinity. Such skillful manipulation is often necessary in higher-level exams like the JEE Advanced, where time management and quick problem-solving abilities are tested.

Greatest Integer Function

The greatest integer function, often denoted by \( [x] \), is an essential concept in discrete mathematics. It returns the largest integer less than or equal to a given real number 'x'. For example, \( [3.7] = 3 \) and \( [-1.2] = -2 \). This step function has a staircase-like graph and is sometimes referred to as the floor function.

This function is particularly interesting because it introduces a form of discontinuity where simple continuity does not apply. It can lead to intriguing limits and series problems where students must carefully consider the properties of integers, fractions, and real numbers, particularly in preparatory exams like the JEE Advanced, where such functions often appear.

Infinity in Limits

Infinity, symbolized by \( \infty \) in mathematics, is a concept describing something without any bound or larger than any natural number. When it comes to limits, the idea of infinity is used to describe the behavior of functions or sequences as they grow larger and larger without any prescribed ending or bound.

Dealing with limits at infinity requires understanding how functions behave as the variable grows without restriction, which often simplifies the function's behavior by diminishing the influence of constants or smaller order terms. As such, the concept of infinity is not just about large numbers, but about the trends and tendencies of mathematical expressions, a critical concept often tested in the JEE Advanced mathematics section.

JEE Advanced Mathematics

The Joint Entrance Examination (JEE) Advanced is an academic challenge that tests a student's proficiency in various subjects, including mathematics, physics, and chemistry, for entrance into engineering programs in India. In the mathematics section, students face complex problems involving algebra, calculus, trigonometry, and more.

Concepts like limit definition, simplifying expressions, greatest integer functions, and handling infinity in limits are crucial for success in this exam. These topics require a deep understanding and the ability to apply concepts to solve intricate problems quickly. Students must have a strong grasp on these fundamental ideas, as they often interconnect and form the basis for more complicated mathematical theorems and applications in the JEE Advanced syllabus.