<p>In the expansion of \(\left[(1+x)/(1-x)\right]^2\), the coefficient of \(x^n\) will be</p>
Step-by-Step Solution
Key Concept: Rewrite (1+x)/(1-x) as (1+x)(1-x)^(-1) and use binomial expansions for both factors, then find the coefficient of x^n by convolution of series.
<p><strong>Step 1:</strong> Write the expression as a product: $\left[\frac{1+x}{1-x}\right]^2 = (1+x)^2 \cdot (1-x)^{-2}$</p><p><strong>Step 2:</strong> Expand $(1+x)^2 = 1 + 2x + x^2$</p><p><strong>Step 3:</strong> Expand $(1-x)^{-2}$ using binomial series: $(1-x)^{-2} = \sum_{k=0}^{\infty} \binom{-2}{k}(-x)^k = \sum_{k=0}^{\infty} (k+1)x^k = 1 + 2x + 3x^2 + 4x^3 + \ldots + (n+1)x^n + \ldots$</p><p><strong>Step 4:</strong> Multiply the series and collect coefficient of $x^n$:</p><p>Coefficient of $x^n$ = $1 \cdot (n+1) + 2 \cdot n + 1 \cdot (n-1)$</p><p>= $(n+1) + 2n + (n-1) = 4n$</p><p>∴ Answer: <strong>4n</strong> (Option A)</p>
Correct Answer: A