Problem 122 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|x|<1\), then the coefficient of \(x^{n}\) in expansion of \((1\) \(\left.+x+x^{2}+x^{3}+\ldots\right)^{2}\) is: (A) \(n\) (B) \(n-1\) (C) \(n+2\) (D) \(n+1\)

Step-by-Step Solution

Answer

The coefficient is \(n+1\).

Step 1: Understand the geometric series

The expression \(1 + x + x^2 + x^3 + \ldots\) represents an infinite geometric series with a common ratio of \(x\). For \(|x| < 1\), this series can be expressed as \(\frac{1}{1-x}\) according to the formula for the sum of an infinite geometric series.

Step 2: Express the problem as a square

The problem asks for the coefficient of \(x^n\) in the expansion of \((1 + x + x^2 + x^3 + \ldots)^2\). Given the series sums to \(\frac{1}{1-x}\), the expression becomes \(\left(\frac{1}{1-x}\right)^2\).

Step 3: Expand \(\left(\frac{1}{1-x}\right)^2\) using binomial series

The binomial series expansion for \(\left(\frac{1}{1-x}\right)^2\) is given by \(\sum_{k=0}^\infty (k+1)x^k\). In this series, each term has a coefficient equal to \(k+1\), corresponding to \(x^k\).

Step 4: Identify the coefficient of \(x^n\)

From the binomial expansion, the term corresponding to \(x^n\) is \((n+1)x^n\). Thus, the coefficient of \(x^n\) is \(n+1\).

Key Concepts

Binomial Series Expansion

The concept of binomial series expansion is a powerful tool used in mathematics, especially when dealing with infinite series. It starts with the traditional binomial theorem, which provides a method to expand expressions of the form \((1 + x)^n\). This expansion results in a polynomial expression with coefficients given by the binomial coefficients. In contexts like our exercise where the expression is more complex, such as \(\left(\frac{1}{1-x}\right)^2\), an adaptation of the binomial series is used.

In such cases, we express the expansion as an infinite series, \(\sum_{k=0}^\infty (k+1)x^k\). Each term in this series has a coefficient that aligns with the modified binomial coefficient \(k+1\). Understanding this expansion allows us to determine specific coefficients for terms like \(x^n\), which is central to solving these sorts of problems.

• Binomial series expansion extends the traditional binomial theorem to infinite series.
• Coefficients are determined by modified binomial expressions such as \(k+1\) in certain series.

Infinite Series

An infinite series sums an endless sequence of terms. It's a critical concept when dealing with expressions like \(1 + x + x^2 + x^3 + \ldots\). This pattern forms an infinite geometric series, especially when \(|x| < 1\). Such a series can be summed using the formula for a geometric series, \(\frac{1}{1-x}\).

Understanding how to identify and sum an infinite series is essential in various mathematical contexts. It simplifies complex expressions and provides a way to find coefficients within series expressions efficiently. The convergence (or validity) of these series is bound by conditions like \(|x| < 1\), ensuring the series eventually leads to a valid summation.

• Infinite series are sums of endless sequences.
• Geometric series can be summed using specific formulas, given certain conditions are met (e.g., \(|x| < 1\)).

Coefficient of Polynomial Terms

In polynomial expressions, coefficients play a crucial role. They are the multipliers of the variable terms and reflect the influence each variable power has on the overall expression. When expanding an expression using a series, such as in our exercise \((1 + x + x^2 + x^3 + \ldots)^2\), identifying these coefficients is critical.

The task involves finding the coefficient of specific terms, like \(x^n\). For the series expansion discussed, each large power term is associated with a coefficient as outlined by a series expansion like \(\sum_{k=0}^\infty (k+1)x^k\).

Understanding how to extract these coefficients helps in identifying patterns and solving for specific values, which are crucial for solving many polynomial and series-related problems.

• Coefficients indicate the importance of variable powers in polynomials.
• Extraction from series expansions aids in identifying term-specific elements like \(x^n\).