<p>The formula \((a + b)^m = a^m + ma^{m-1}b + \dfrac{m(m+1)}{1 \cdot 2} a^{m-2}b^2 + \cdots\) holds when</p>
Step-by-Step Solution
Key Concept: The binomial series expansion (a+b)^m converges only when |b/a| < 1, which ensures the infinite series terminates or converges. This requires recognizing that the formula shown is the generalized binomial series, not the finite binomial theorem.
<p><strong>Step 1:</strong> Identify the formula type. The given formula with terms like m(m+1)/2 in binomial coefficients is the <strong>generalized binomial series</strong>, not the finite binomial theorem.</p><p><strong>Step 2:</strong> The generalized binomial expansion (a+b)^m = a^m[1 + (b/a)]^m is valid when the series converges.</p><p><strong>Step 3:</strong> Using the binomial series for (1+x)^m, convergence requires |x| < 1. Here x = b/a, so we need |b/a| < 1, or equivalently <strong>|b| < |a|</strong>.</p><p><strong>Step 4:</strong> Alternatively, this can be stated as |a| > |b| or |a/b| > 1.</p><p>∴ Answer: D (The condition is |a| > |b| or |b/a| < 1)</p>
Correct Answer: D