<p>Find the sum of infinite series
\[\frac{1}{1\times3\times5}+\frac{1}{3\times5\times7}+\frac{1}{5\times7\times9}+\cdots\]</p>
Step-by-Step Solution
Key Concept: Use partial fraction decomposition to express each term as a difference of two fractions, creating a telescoping series where consecutive terms cancel.
<p><strong>Step 1: Set up partial fractions</strong></p><p>For the general term, let n = 1,2,3,... so terms are (2n-1)(2n+1)(2n+3).</p><p>Decompose: <strong>1/[(2n-1)(2n+1)(2n+3)] = (1/4)[1/((2n-1)(2n+1)) - 1/((2n+1)(2n+3))]</strong></p><p>Verify: RHS = (1/4)[(2n+3) - (2n-1)]/[(2n-1)(2n+1)(2n+3)] = (1/4)[4]/[(2n-1)(2n+1)(2n+3)] ✓</p><p><strong>Step 2: Write out the telescoping series</strong></p><p>S = (1/4)[1/(1×3) - 1/(3×5)] + (1/4)[1/(3×5) - 1/(5×7)] + (1/4)[1/(5×7) - 1/(7×9)] + ...</p><p><strong>Step 3: Apply telescoping</strong></p><p>S = (1/4)[1/(1×3) - lim(n→∞) 1/((2n+1)(2n+3))]</p><p>The limit term vanishes as n→∞.</p><p><strong>Step 4: Calculate final answer</strong></p><p>S = (1/4) × 1/3 = <strong>1/12</strong></p>
Correct Answer: 1/12