Problem 8 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\sum_{n=1}^{\infty} x^{n-1}=a\) and \(\sum_{n=1}^{\infty} y^{n-1}=b\) where \(|x|,|y|<1\), then \(\sum_{n=1}^{\infty}(x y)^{n-1}=\) (A) \(a b\) (B) \(\frac{a+b-1}{a b}\) (C) \(\frac{1}{1-a b}\) (D) \(\frac{a b}{a+b-1}\)

Step-by-Step Solution

Answer

(C) \(\frac{1}{1-ab}\)

Step 1: Identify the Given Series

The series given are \( \sum_{n=1}^{\infty} x^{n-1}=a \) and \( \sum_{n=1}^{\infty} y^{n-1}=b \). These are geometric series with first term \( 1 \) and common ratios \( x \) and \( y \), respectively.

Step 2: Recall the Formula for a Geometric Series

The sum of an infinite geometric series \( \sum_{n=0}^{\infty} r^{n} \) is \( \frac{1}{1-r} \), provided that \( |r|<1 \). The given series \( \sum_{n=1}^{\infty} x^{n-1} \) can be rewritten as \( \sum_{n=0}^{\infty} x^{n} \) which simplifies to \( \frac{1}{1-x} \). Similarly, for \( y \), it is \( \frac{1}{1-y} \).

Key Concepts

Infinite Series

An infinite series is a sum of infinitely many terms. It continues without end or limit.
In mathematics, we often deal with infinite series to understand patterns or solve complex problems.
In the exercise, we encounter infinite series of the form \[ \sum_{n=1}^{\infty} x^{n-1} = a \] and \[ \sum_{n=1}^{\infty} y^{n-1} = b \].
These represent geometric series that include infinite terms starting from the power of zero.
The infinite nature of such series makes them valuable for precise calculations in various fields.

Common Ratio

The concept of a common ratio is key to understanding geometric series.
In any geometric series, the common ratio is the constant factor between consecutive terms.
If the terms of a series are: 1, 3, 9, 27, ..., the common ratio is 3 because each term is 3 times the previous one.For our given series:

• The common ratio for \(\sum_{n=1}^{\infty} x^{n-1}=a\) is \x\.
• The common ratio for \(\sum_{n=1}^{\infty} y^{n-1}=b\) is \y\.

Understanding the common ratio helps us to evaluate the infinite series and apply the sum formula correctly.

Convergence of Series

Convergence of a series refers to whether the series approaches a finite value as more terms are added.
A series converges if adding more terms results in a sum that stabilizes to a particular number.In the case of geometric series, the series will converge if the absolute value of the common ratio is less than one.
Mathematically, a geometric series \(\sum_{n=0}^{\infty} r^{n}\) converges to \(\frac{1}{1-r}\) if \(|r| < 1\).

• In our exercise, both \(|x| < 1\) and \(|y| < 1\), ensuring that both series are convergent.
• This property is crucial, as it allows us to represent and manipulate these infinite sums neatly as \(\frac{1}{1-x}\) and \(\frac{1}{1-y}\), respectively.

Convergence is a foundational concept when dealing with infinite series, ensuring they can be used in valid mathematical operations.