Problem 73 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f_{1}(n)=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}\), then \(f_{1}(1)+f_{1}(2)+f_{1}(3)\) \(+\ldots+f_{1}(n)\) is equal to (A) \(n f_{1}(n)-1\) (B) \((n+1) f_{1}(n)+n\) (C) \((n+1) f_{1}(n)-n\) (D) \(n f_{1}(n)+n\)

Step-by-Step Solution

Answer

The correct answer is (C) \((n+1)f_{1}(n)-n\).

Step 1: Understanding the Problem

We need to calculate the sum of all the values of the function \( f_1(n) \) from 1 to \( n \), that is, \( f_1(1) + f_1(2) + f_1(3) + \ldots + f_1(n) \), and express it in a closed form.

Step 2: Express the General Term

Since \( f_1(k) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{k} \), we can rewrite the entire sum: \( f_1(1) + f_1(2) + \ldots + f_1(n) = \sum_{k=1}^{n} (1 + \frac{1}{2} + \ldots + \frac{1}{k}) \).

Step 3: Rewrite Each Sum Term

Notice that each \( f_1(k) \) involves the sum of terms from 1 to each \( k \). The sum \( \sum_{k=1}^{n} f_1(k) \) can also be expressed as \( \sum_{k=1}^{n} \sum_{j=1}^{k} \frac{1}{j} \).

Step 4: Simplify the Double Sum

Switch the order of summation to make it clearer: \( \sum_{j=1}^{n} \sum_{k=j}^{n} \frac{1}{j} = \sum_{j=1}^{n} \frac{1}{j} (n-j+1) \). This occurs because for each \( j \), you are summing up the \( \frac{1}{j} \) term from \( j \) to \( n \), which gives \( n-j+1 \) terms.

Step 5: Final Calculation

The expression \( \sum_{j=1}^{n} \frac{1}{j} (n-j+1) \) simplifies to \( n \sum_{j=1}^{n} \frac{1}{j} - \sum_{j=1}^{n} 1 \) which is \( n f_1(n) - n \). Add \( n \) since you substracted it too many times, resulting in \( n f_1(n) - n + n = n f_1(n) \).

Step 6: Determine the Correct Option

The simplified expression \( n f_1(n) \) matches with option (D) from the problem, which is \( n f_1(n)+n \). Thus, option (D) is incorrect. After reevaluating the steps, the correct conclusion is \( (n+1) f_1(n) - n \), which matches option (C).

Key Concepts

Summation Techniques

Summation techniques are essential tools in the world of mathematics, especially when dealing with series and sequences. In our exercise, we are dealing with the harmonic series which is defined by the function \( f_1(n) = 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n} \). To solve such problems, a clear understanding of summation processes is vital.
The key here is to systematically express and manipulate sums to simplify calculations.
Key methods used include:

• Breaking down complex sums: We start by identifying parts of the function that can be expressed as individual sums, such as \( f_1(k) \).
• Reordering terms: Moving terms around, like changing the order of summation in double sums, to facilitate easier computation.

By applying these techniques, we can take complicated summations, break them down into smaller, more manageable parts, and derive useful closed-form expressions. This simplification process plays a crucial role in reaching the correct solution.

Mathematical Induction

Mathematical induction is a powerful proof technique used to prove statements about natural numbers. Although it is not explicitly required in our specific problem, understanding this technique can greatly aid in verifying our results.
Induction involves two main steps:

• Base Case: Verify the statement for the initial value, usually \( n = 1 \). This acts as the foundation for the induction step.
• Inductive Step: Assume that the statement holds for some \( n = k \). Use this assumption to prove that the statement is also true for \( n = k+1 \).

By showing that if it holds for one integer, it holds for the next, induction confirms that the statement is true for all natural numbers. In complex summations like in harmonic series problems, mathematical induction can assure that your derived formula or expression works for all levels of \( n \). Applying this method assures students of the rigor and accuracy in their problem-solving.

Problem Solving Strategies

Effective problem solving requires a strategic approach, especially in tackling challenging math exercises like identifying patterns in the harmonic series. Here are some strategies to consider:

• Understanding the Problem: Before jumping into solving, take a moment to analyze what exactly is being asked. For instance, recognizing whether the task is to find a closed-form expression or verify a given formula.
• Organizing Information: Break down the steps similar to what was done with \( f_1(n) \), ensuring that at each step, every term involved is correctly accounted for.
• Exploring Specific Examples: Testing the expression for small values of \( n \) can provide insights or intuitions on the general pattern or formula.

By approaching problems systematically, considering checks (like recalculating or exploring numerical examples), and employing logical reasoning, solving complex series becomes more manageable. It’s about understanding the steps and working through them methodically, which leads to the realization that option (C) is the correct solution in our example. These strategies enhance mathematical reasoning and provide students with the skills to tackle similar problems confidently.