<p>If \(|x| < 1\), then the coefficient of \(x^n\) in expansion of \((1 + x + x^2 + x^3 + \cdots)^2\) is</p>
Step-by-Step Solution
Key Concept: When |x| < 1, the binomial series (1+x)^n converges to an infinite series. For non-integer n, this expansion differs fundamentally from the finite binomial theorem—you must use the generalized binomial coefficient C(n,r) = n(n-1)...(n-r+1)/r! and sum infinitely many terms.
<p><strong>Step 1:</strong> Recognize this requires the <strong>generalized binomial theorem</strong> (not the finite one), since n is not a positive integer and |x| < 1 ensures convergence.</p><p><strong>Step 2:</strong> For |x| < 1, the expansion is:
(1+x)^n = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + ... (infinite series)</p><p><strong>Step 3:</strong> The general term (r-th term, r ≥ 0) is:
T_r = C(n,r)x^r where C(n,r) = n(n-1)(n-2)...(n-r+1)/r!</p><p><strong>Step 4:</strong> The series converges for |x| < 1 and diverges for |x| ≥ 1.</p><p>∴ Answer: D</p>
Correct Answer: D