<p>\(1 + \dfrac{1}{4} + \dfrac{1 \times 3}{4 \times 8} + \dfrac{1 \times 3 \times 5}{4 \times 8 \times 12} + \cdots =\)</p>
Step-by-Step Solution
Key Concept: Recognize this series as the binomial expansion of (1+x)^n where the general term matches the pattern from (1+x)^(1/2). Use the coefficient formula for fractional powers: the nth term involves products of odd numbers in numerator and multiples of 4 in denominator.
<p><strong>Step 1:</strong> Identify the general term pattern. The nth term (starting n=0) is: [1·3·5·...·(2n-1)]/[4·8·12·...·4(n+1)]</p><p><strong>Step 2:</strong> Rewrite as: [1·3·5·...·(2n-1)]/[(2^2)·(2^3)·(2^4)·...·(2^(n+1))] = [1·3·5·...·(2n-1)]/[2^(2+3+4+...+(n+1))]</p><p><strong>Step 3:</strong> Recognize this matches the binomial coefficient: C(1/2, n) = [1·3·5·...·(2n-1)]/[n!·2^n]</p><p><strong>Step 4:</strong> The series is: Σ C(1/2, n)·(1/4)^n = (1 + 1/4)^(1/2) using the binomial theorem (1+x)^n = Σ C(n,k)x^k</p><p><strong>Step 5:</strong> Evaluate: (5/4)^(1/2) = √(5/4) = √5/2</p><p>∴ Answer: A</p>
Correct Answer: A