Problem 115 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(f, g\) and \(h\) be real valued functions defined on an interval \(I \subseteq R\) except possibly for some point \(c\) such that $$ \lim _{x \rightarrow c} f(x)=l=\lim _{x \rightarrow c} h(x) $$ and, \(f(x) \leq g(x) \leq h(x), \forall x \in I .\) Then, \(\lim _{x \rightarrow c} g(x)=l\). \(\lim _{n \rightarrow \infty} \frac{\\{x\\}+\\{2 x\\}+\\{3 x\\}+\ldots+\\{n x\\}}{n^{2}}\), where \(\\{x\\}=x-[x]\) denotes the fractional part of \(x\), is (A) 1 (B) 0 (C) \(\frac{1}{2}\) (D) None of these

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Fractional Part

The fractional part of a real number \( x \), denoted as \( \{ x \} \), is the part of \( x \) that remains after subtracting the integer part. Mathematically, \( \{ x \} = x - \lfloor x \rfloor \). This means \( 0 \leq \{ x \} < 1 \) for any real number \( x \).

Step 2: Expression of the Problem

The problem requires us to compute \( \lim_{n \to \infty} \frac{\{x\}+\{2x\}+\{3x\} + \ldots + \{nx\}}{n^2} \). Given that each \( \{kx\} \) is less than 1, \( \{x\} + \{2x\} + \ldots + \{nx\} \) is at most \( n \).

Step 3: Establishing Bounds for the Series

Because each term \( \{kx\} \) in the series is between 0 and 1, the sum \( S_n = \{x\} + \{2x\} + \ldots + \{nx\} \) satisfies \( 0 \leq S_n \leq n \). This gives us the inequality \( 0 \leq \frac{S_n}{n^2} \leq \frac{n}{n^2} = \frac{1}{n} \).

Step 4: Taking the Limit

Given the bounds from the previous step, for all natural \( n \), \( \frac{S_n}{n^2} \leq \frac{1}{n} \). As \( n \to \infty \), \( \frac{1}{n} \to 0 \). Thus, by the Squeeze Theorem, \( \lim_{n \to \infty} \frac{S_n}{n^2} = 0 \).

Key Concepts

Fractional Part

The fractional part of a real number is a simple yet important concept in mathematics. This part helps us understand what remains of a number after we remove its integer component. For a real number \( x \), the fractional part, denoted as \( \{ x \} \), is calculated by subtracting the floor of \( x \) from \( x \). Mathematically, this is represented as \( \{ x \} = x - \lfloor x \rfloor \). Here, \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \).

• The fractional part is always non-negative.
• It lies within the interval \([0, 1)\).

Understanding this concept is crucial for solving problems involving sequences where terms are bounded. For instance, when we look at terms like \( \{ kx \} \) for different integer values of \( k \), we can harness these properties to establish bounds or limits.

Limit of a Sequence

In mathematical analysis, the limit of a sequence is a concept that describes the value a sequence approaches as the index becomes infinitely large. When we say a sequence \( a_n \) has a limit \( L \) as \( n \to \infty \), it means that the terms of the sequence get arbitrarily close to \( L \) for sufficiently large \( n \).

To formally define, \( \lim_{n \to \infty} a_n = L \) if for every \( \epsilon > 0 \), there exists a natural number \( N \) such that for all \( n > N \), \( |a_n - L| < \epsilon \). This definition captures the idea of being able to "squeeze" the sequence's terms closer to \( L \) as \( n \) increases.

In our exercise, we wanted to find \( \lim_{n \to \infty} \frac{S_n}{n^2} \), where \( S_n = \{x\} + \{2x\} + \ldots + \{nx\} \). By applying the Squeeze Theorem and bounding \( \frac{S_n}{n^2} \) between 0 and \( \frac{1}{n} \), we concluded that the limit is 0. This illustrates how bounding helps in determining limits, leveraging the understanding that as sequences grow, their behavior resembles that of determining function limits at infinity.

Inequality in Real Analysis

Inequalities play a pivotal role in real analysis, serving as tools for establishing bounds and relating different functions or sequences. In our current context, we use inequalities to express bounds on sums of fractional parts. The key inequality derived in our exercise was \( 0 \leq S_n \leq n \), where \( S_n \) is the sum of fractional parts.

Real analysis leverages inequalities to apply the Squeeze Theorem, helping us to evaluate limits effectively. The Squeeze Theorem states that if \( a_n \leq b_n \leq c_n \) for all \( n \), and \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} c_n = L \), then \( \lim_{n \to \infty} b_n = L \) as well.

• Inequalities are essential for comparing terms within a sequence.
• They help establish bounds, enabling the use of powerful theorems like the Squeeze Theorem.

Within our specific problem, inequalities helped us understand the behavior of \( \frac{S_n}{n^2} \) as \( n \) grows, allowing us to effectively squeeze and thus determine the limit to be zero, using the understanding of bounds and convergence.