Problem 71 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For a positive integer \(n\), let \(a(n)=\) \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{\left(2^{n}\right)-1} .\) Then (A) \(a(100) \leq 100\) (B) \(a(100)>100\) (C) \(a(200) \leq 100\) (D) \(a(200)>100\)

Step-by-Step Solution

Answer

(A) and (D) are correct: \(a(100) \leq 100\), \(a(200) > 100\).

Step 1: Understanding the Series

The function \( a(n) \) is defined as the sum of the series \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{(2^n) - 1} \). This is a truncated harmonic series, which is known to grow logarithmically.

Step 2: Approximation of the Harmonic Series

The harmonic series \( H_k = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{k} \) has an approximate value of \( \log(k) + \gamma \), where \( \gamma \) is the Euler-Mascheroni constant (approximately 0.577).

Step 3: Applying the Approximation

For \( n = 100 \), the series terminates at \( k = 2^{100} - 1 \). Using the approximation, \( a(100) \approx \log(2^{100}) + \gamma = 100\log(2) + \gamma \). Since \( \log(2) \approx 0.693 \), we have \( 100 \times 0.693 + 0.577 \approx 69.9 + 0.577 = 70.477 \).

Step 4: Comparison for n = 100

As calculated, \( a(100) \approx 70.477 \), which is much less than 100, confirming that option (A) holds: \( a(100) \leq 100 \).

Step 5: Recalculation for n = 200

For \( n = 200 \), the series truncates at \( k = 2^{200} - 1 \). Using the same approximation again, \( a(200) \approx \log(2^{200}) + \gamma = 200\log(2) + \gamma \). Simplifying gives \( 200 \times 0.693 + 0.577 \approx 138.2 + 0.577 = 138.777 \).

Step 6: Comparison for n = 200

The calculation shows \( a(200) \approx 138.777 \), which is greater than 100. Therefore, option (D) holds: \( a(200) > 100 \).

Key Concepts

Harmonic Series

The harmonic series is a fascinating mathematical concept, often encountered in calculus and number theory. It is defined as the sum of the reciprocals of the natural numbers:

• 1 + \( \frac{1}{2} \) + \( \frac{1}{3} \) + \( \frac{1}{4} \) + \ldots

A key characteristic of the harmonic series is that it diverges, meaning it grows without bound as more terms are added. However, it does so quite slowly. This is why, despite the illusion of an eventual sum limit, extending the series to infinity will continue to increase its total value.
In exercises involving the harmonic series, like the truncated version seen in the problem, only a finite number of terms are considered. This approach partially tames the series but also retains its intriguing properties. Truncated harmonic series are used in various mathematical approximations and practical problem-solving scenarios.

Logarithmic Growth

Logarithmic growth describes the slow increase in the value of a series or function compared to linear or exponential growth. In mathematical terms, as the variable in a logarithmic function increases, the overall rate of increase slows down.
For the harmonic series, this attribute is particularly interesting because while the series itself diverges, its growth can be approximated logarithmically. The harmonic series' sum of the first \( n \) terms is roughly equal to \( \log(n) + \gamma \), where \( \gamma \) is the Euler-Mascheroni constant.
This slow expansion resembles logarithmic functions, which is why the harmonic series is often discussed in terms of logarithmic growth. These properties allow for practical predictions in computational mathematics and aid in understanding the nature of infinity in series.
Understanding logarithmic growth is critical in comparing different types of series and functions, revealing insights into their long-term behaviors and relationships.

Euler-Mascheroni Constant

The Euler-Mascheroni constant, denoted by \( \gamma \), is a significant constant in mathematics, appearing in various realms such as number theory, calculus, and complex analysis. Its approximate value is 0.57721566.
The constant emerges in the formula used to approximate the harmonic series: \( H_k \approx \log(k) + \gamma \). This relationship highlights \( \gamma \)'s importance in understanding the deviation between harmonic series sums and their logarithmic counterparts.

• \( \gamma \) provides a bridge between linear and logarithmic growth trends.
• It aids in refining estimates that involve divergent series or integrals.

Though \( \gamma \)'s exact properties remain one of the great unsolved mysteries of mathematics, its utility is undeniable.
Whether you're dealing with mathematical theorems, or the intricate details of integrals, the Euler-Mascheroni constant proves invaluable. Understanding this constant helps in approximating series and understanding limits more accurately.
Overall, \( \gamma \) is a cornerstone in the study of series, playing a vital role particularly when exploring the nuances of growth in mathematical sequences.