Problem 23 - Algebra for JEE ADVANCED

Question

If \(|a|<1\) and \(|b|<1\), then the sum of the series \(1+(1+a) b\) \(+\left(1+a+a^{2}\right) b^{2}+\left(1+a+a^{2}+a^{3}\right) b^{3}+\cdots\) is a. \(\frac{1}{(1-a)(1-b)}\) b. \(\frac{1}{(1-a)(1-a b)}\) c. \(\frac{1}{(1-b)(1-a b)}\) d. \(\frac{1}{(1-a)(1-b)(1-a b)}\)

Step-by-Step Solution

Answer

The sum of the series is option d: \(\frac{1}{(1-a)(1-b)(1-ab)}\).

Step 1: Understand the Series

Given the series: \(1 + (1+a) b + (1+a+a^2) b^2 + (1+a+a^2+a^3) b^3 + \cdots\). Notice that each term is of the form \((1 + a + a^2 + \cdots + a^n)b^n\) for \(n \geq 0\). This is a sum of geometric series within another series.

Step 2: Recognize Inner Geometric Series

The inner series \((1 + a + a^2 + \cdots + a^n)\) is a finite geometric series. The formula for the sum of the first \(n+1\) terms of a geometric series is \(\frac{1-a^{n+1}}{1-a}\) where \(|a| < 1\).

Step 3: Express Each Term and Simplify

Each term becomes \(\frac{1-a^{n+1}}{1-a}b^n\). So the series can be rewritten as: \[\frac{1}{1-a} + \frac{1-a^2}{1-a}b + \frac{1-a^3}{1-a}b^2 + \frac{1-a^4}{1-a}b^3 + \cdots\]

Step 4: Identify the Overall Series Pattern

Recognize the series as a sum of separate series: \(\frac{1}{1-a}(1 + b + b^2 + b^3 + \cdots)\) and a correction term as a series of \(a^n b^n\). This turns into a geometric series \(\sum b^n = \frac{1}{1-b}\) and a second geometric series \(\sum (ab)^n = \frac{1}{1-ab}\).

Step 5: Combine the Results

Combine the series results: The total sum can be computed by multiplying the geometric series solutions: \(\frac{1}{1-a} \times \frac{1}{1-b}\) taking into account that \(a\) and \(b\) are within their respective convergences. Correct by dividing by \(1-ab\) to make the separate corrections fit, resulting in \(\frac{1}{(1-a)(1-b)(1-ab)}\).

Step 6: Match with Options

The derived formula matches with option d: \(\frac{1}{(1-a)(1-b)(1-ab)}\). The correct choice that encompasses the behavior of both inner and outer series as separate yet combined through corrections is option d.

Key Concepts

Geometric Series

A geometric series is a special type of series where each term is obtained by multiplying the previous term by a constant. This constant is known as the common ratio. For example, in a geometric series such as \(1, r, r^2, r^3, \ldots\), the common ratio is \(r\).
A geometric series can be finite or infinite. The sum of the first \(n\) terms of a finite geometric series is given by:

• \( S_n = \frac{1-r^{n+1}}{1-r} \)

where \(|r| < 1\). This formula allows us to calculate the sum efficiently. In the given exercise, each sub-series, like \(1+a+a^2+\ldots\), is a finite geometric series. Recognizing this allows one to simplify calculations and find the total sum of the series more easily.

Convergence

Convergence refers to the behavior of a series as the number of terms increases indefinitely. It determines whether a series approaches a finite value or diverges to infinity. For a geometric series, convergence is determined by the common ratio \(r\).
If \(|r| < 1\), the infinite geometric series converges, meaning it approaches a specific value. The sum of an infinite geometric series is:

• \( S = \frac{1}{1-r} \)

In the exercise, convergence is crucial because both \(a\) and \(b\) satisfy \(|a| < 1\) and \(|b| < 1\). Ensuring convergence helps achieve a finite sum for the series.

Series Summation

Series summation involves determining the total sum of all terms within a series. It is an essential process in working with both finite and infinite series. For a geometric series, once you recognize the type and understand its pattern, using the summation formulas becomes straightforward.
In the exercise, each term can be broken down into parts that resemble known series. By identifying the series as geometric, applying the summation formulas for both finite and infinite cases becomes easy. Combining these results, like finding the sum for inner \((1 + a + a^2 + \ldots)\) and outer \(b^n\), is key to solving such complex series.

Infinite Series

An infinite series is a series that continues indefinitely without terminating. The sum of an infinite series depends on the convergence of the series, as not all infinite series have a finite sum.
For an infinite geometric series to have a sum, the absolute value of the common ratio must be less than one.
In the exercise, the concept of an infinite series comes into play, because the terms continue indefinitely as powers of \(b\) and \(ab\). Recognizing where the infinite series appear and determining their convergence allows us to add them up correctly, using their geometric sum formulas. These considerations are foundational when dealing with series that appear to extend to infinity but still offer meaningful and finite sums.