Problem 92 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(H_{n}=1+\frac{1}{2}+\frac{1}{3}+\ldots .+\frac{1}{n}\), then the value of \(1+\frac{3}{2}+\frac{5}{3}+\ldots+\frac{2 n-1}{n}\) is (A) \(n-H_{n}\) (B) \(2 n-H_{n}\) (C) \((n-1)-H_{n}\) (D) \(n-2 H_{n}\)

Step-by-Step Solution

Answer

The value is \(2n - H_n\), which is option (B).

Step 1: Understanding the Problem

The problem involves calculating the sum of a series: \( S = 1 + \frac{3}{2} + \frac{5}{3} + \ldots + \frac{2n-1}{n} \). We need to identify a relation between this series and the harmonic series defined by \( H_n = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} \).

Step 2: Rewrite the Series

Write the terms of the series as \( \frac{1}{1} + \frac{3}{2} + \frac{5}{3} + \ldots + \frac{2n-1}{n} \). Generally, each term can be expressed as \( \frac{2k-1}{k} \) for \( k = 1, 2, 3, \ldots, n \).

Step 3: Separate Each Term

Each term \( \frac{2k-1}{k} \) can be split into two fractions: \( 2 - \frac{1}{k} \). Therefore, the series \( S = (2 - 1) + (2 - \frac{1}{2}) + (2 - \frac{1}{3}) + \ldots + (2 - \frac{1}{n}) \).

Step 4: Simplify the Series

Simplifying the above expression gives \( S = \underbrace{2+2+2+...+2}_{n \text{ times}} \ - \left(1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}\right) \). This results in \( S = 2n - H_n \).

Step 5: Interpretation and Conclusion

By comparing the expression \( S = 2n - H_n \) with the given options, we see that option (B) matches the derived expression for our series \( S \).

Key Concepts

Series Summation

The concept of series summation involves adding up a sequence of numbers to find their total sum. In our problem, we are tasked with summing the series that looks like this:

• \( S = 1 + \frac{3}{2} + \frac{5}{3} + \ldots + \frac{2n-1}{n} \)

Series summation is essential in mathematics because it allows us to sum an infinite or finite set of numbers. Here, each term of the series is given by \( \frac{2k-1}{k} \) where \( k \) runs from 1 to \( n \). This setup helps us utilize known series like the harmonic series, and break it down further into simpler parts for computation. For instance, we separate the terms into two parts: one being a constant term \( 2 \), and the other being the reciprocal part \( \frac{1}{k} \). This allows us to express our series sum as combinations of simpler known sums like the sum of constant values and harmonic series terms.

Mathematical Sequences

In mathematics, a sequence is a set of numbers that follow a particular pattern. Each number in a sequence is referred to as a term. Within our exercise, we're dealing with the arithmetic sequence defined by the terms \( \frac{2k-1}{k} \).

• An arithmetic sequence has a specific pattern in which each term is formed by adding a constant difference to the preceding term.
• In the context of our problem, each numerator \((2k-1)\) depicts an arithmetic sequence of odd numbers, while the denominator is a simple progression from 1 to \( n \).
• Understanding this setup allows us to manipulate these terms into smaller, simpler sequences. For instance, splitting \( \frac{2k-1}{k} \) into \( 2 - \frac{1}{k} \) simplifies the arithmetic operations needed when summing.

This concept is a cornerstone in higher mathematics as it allows manipulation and simplification, which is especially useful in problems like those found in competitive examinations.

JEE Mathematics

The Joint Entrance Examination (JEE) is one of the most competitive engineering entrance assessments in India. As part of its syllabus, mastering series and sequences, like the harmonic and arithmetic series, is crucial.

• Skills in manipulating series and sequences are key, allowing students to tinker with numbers to find manageable solutions.
• Problems, such as the one we solved, often appear in JEE exams to test a student's understanding of combining sequences and series.
• Fluency with sequences boosts problem-solving speed by recognizing patterns and breaking down complex expressions efficiently.

By working through these types of problems, students are able to enhance their mathematical thinking and apply these concepts to various scenarios not only in exams but in real-world situations where mathematical modeling is involved.