<p>Sum the series: \[\frac{7}{5}\left(1 + \frac{1}{10^2} + \frac{1\cdot3}{1\cdot2}\cdot\frac{1}{10^4} + \frac{1\cdot3\cdot5}{1\cdot2\cdot3}\cdot\frac{1}{10^6} + \cdots \text{ to } \infty\right).\]</p>
Step-by-Step Solution
Key Concept: Recognize the series inside the parentheses as the binomial expansion of (1+x)^(-1/2) with x = 1/100, using the generalized binomial theorem for fractional exponents.
<p><strong>Step 1:</strong> Identify the series pattern. The general term is: 1·3·5·...·(2n-1) / (1·2·3·...·n) · (1/10^(2n)) = C(-1/2, n) · (1/100)^n where C(-1/2, n) is the generalized binomial coefficient.</p><p><strong>Step 2:</strong> Recognize this as the binomial expansion: (1+x)^(-1/2) = Σ C(-1/2, n)·x^n, where x = 1/100.</p><p><strong>Step 3:</strong> Evaluate (1 + 1/100)^(-1/2) = (101/100)^(-1/2) = (100/101)^(1/2) = 10/√101.</p><p><strong>Step 4:</strong> The series inside parentheses equals 10/√101. Therefore: (7/5) · (10/√101) = 70/(5√101) = 14/√101 = 14√101/101.</p><p>∴ Answer: <strong>14√101/101</strong> or equivalently <strong>14/√101</strong></p>
Correct Answer: 14