Problem 130 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the expansion in powers of \(x\) of the function \(\frac{1}{(1-a x)(1-b x)}\) is \(a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\ldots\), then \(\mathrm{a}_{n}\) is (A) \(\frac{b^{n}-a^{n}}{b-a}\) (B) \(\frac{a^{n}-b^{n}}{b-a}\) [2006] (a) \(\frac{a^{n+1}-b^{n+1}}{b-a}\) (D) \(\frac{b^{n+1}-a^{n+1}}{b-a}\)

Step-by-Step Solution

Answer

\( a_n = \frac{a^n - b^n}{b-a} \); choice (B).

Step 1: Express the Function in a Different Form

The given function, \( \frac{1}{(1-a x)(1-b x)} \), can be expanded as a series in powers of \( x \). First, we rewrite it using the partial fraction decomposition: \( \frac{1}{(1-a x)(1-b x)} = \frac{1}{b-a} \left( \frac{1}{1-ax} - \frac{1}{1-bx} \right) \). This allows us to separate the terms into two geometric series.

Step 2: Expand Using Geometric Series

The expression \( \frac{1}{1-ax} \) expands to \( 1 + ax + a^2x^2 + a^3x^3 + \ldots \), and \( \frac{1}{1-bx} \) expands similarly to \( 1 + bx + b^2x^2 + b^3x^3 + \ldots \). These expansions allow us to express the function as a sum of two infinite series.

Step 3: Subtract the Series

When substituting the geometric series into our partial fraction expansion, we have: \( \frac{1}{b-a} \left( (1 + ax + a^2x^2 + \ldots) - (1 + bx + b^2x^2 + \ldots) \right) \). Combining terms with the same power of \( x \), the coefficients become \( \frac{a^n - b^n}{b-a} \), defining the terms \( a_n \).

Step 4: Identify Correct Choice

The result from Step 3 matches choice (B), \( \frac{a^n - b^n}{b-a} \). This confirms the expression for \(a_n\) is correct according to the steps taken to solve the problem.

Key Concepts

Geometric Series

In mathematical analysis, a geometric series is a series with a constant ratio between successive terms. This means that each term in the series is the previous term multiplied by a fixed number, known as the common ratio. For example, the series \( 1 + ar + ar^2 + ar^3 + \ldots \) is geometric because each term is multiplied by \( r \) to obtain the next one.
In the given problem, the expressions \( \frac{1}{1-ax} \) and \( \frac{1}{1-bx} \) are examples of geometric series. The formula for such a series is \( \frac{1}{1-r} = 1 + r + r^2 + r^3 + \ldots \), for \(|r|<1\).
This formula helps in expanding the expressions as a power series, breaking the original function into simpler components that can be individually analyzed. It is crucial to recognize a geometric series, as it significantly simplifies the process of expansion and finding coefficients in problems like this one.

Power Series Expansion

A power series is an infinite sum of terms in the form of \( a_n x^n \), where \( n \) is a non-negative integer, and \( a_n \) denotes the coefficients of the series. Power series expansions are hugely helpful in approximating functions by expressing them as the sum of simpler polynomial terms.
In this particular exercise, after rewriting the function \( \frac{1}{(1-ax)(1-bx)} \) using partial fraction decomposition, we expanded each part into its respective geometric series. By doing so, we transformed the function into a power series in terms of \( x \).
The power series representation is useful for determining the coefficients \( a_n \), which are analyzed to solve the exercise question. This method allows us to focus on each term separately and makes it easier to manipulate the series for finding specific information, such as the coefficient expressions in options (A) to (D).

Problem Solving Steps

Solving complex mathematical problems often requires a structured approach, incorporating clear steps that guide us to the solution methodically. Here is a simplified overview of how we approached this particular problem:

• **Rewrite the Function:** Start by rewriting the given function \( \frac{1}{(1-a x)(1-b x)} \) using partial fraction decomposition. This step breaks the single complex function into simpler fractions that can be more easily managed.


• **Expand into Series:** Recognize the opportunity to use geometric series to expand each fraction separately. We expanded \( \frac{1}{1-ax} \) and \( \frac{1}{1-bx} \) as geometric series, which laid the groundwork for simplifying the problem.


• **Combine and Simplify:** Subtract the resulting series from each other, allowing us to combine terms with the same power of \( x \). This manipulative step reveals the coefficients of the power series we are seeking, specifically \( \frac{a^n - b^n}{b-a} \).


• **Select the Correct Answer:** Finally, by matching our derived solution to the given options, we identify (B) as the correct choice. This confirmation validates our process and solidifies our understanding of the function's power series expansion.

Following structured steps ensures clarity and keeps focus on the strategic breakdown of mathematical complexities, making problem-solving both efficient and effective.