Limits, Continuity & Differentiability
Limits using series expansion
Grade 12
Question:
<p>If \( \lim_{x \to 0} \dfrac{10 - \displaystyle\sum_{k=1}^{10}(\cos kx)}{x^2} = \dfrac{a}{b} \) where <em>a</em> and <em>b</em> are co-prime, then the value of \( (a + b) \) is equal to:</p>
<p>(a) 384</p>
<p>(b) 385</p>
<p>(c) 386</p>
<p>(d) 387</p>
Step-by-Step Solution
Key Concept: Use Taylor expansion of cos(kx) = 1 - (kx)²/2 + O(x⁴) for each term, then sum the quadratic coefficients. The numerator becomes x² times a sum of k² values, which evaluates to 385 for k=1 to 10.
<p><strong>Step 1:</strong> Expand each cosine term using Taylor series:</p><p>cos(kx) = 1 - (kx)²/2 + O(x⁴) = 1 - k²x²/2 + O(x⁴)</p><p><strong>Step 2:</strong> Sum all 10 terms:</p><p>∑_{k=1}^{10} cos(kx) = 10 - x²/2 · ∑_{k=1}^{10} k² + O(x⁴)</p><p><strong>Step 3:</strong> Calculate ∑_{k=1}^{10} k² using the formula n(n+1)(2n+1)/6:</p><p>∑_{k=1}^{10} k² = 10(11)(21)/6 = 385</p><p><strong>Step 4:</strong> Find the numerator:</p><p>10 - ∑_{k=1}^{10} cos(kx) = 10 - [10 - 385x²/2 + O(x⁴)] = 385x²/2 + O(x⁴)</p><p><strong>Step 5:</strong> Evaluate the limit:</p><p>lim_{x→0} (385x²/2)/x² = 385/2</p><p><strong>Step 6:</strong> Since gcd(385, 2) = 1 (385 = 5 × 7 × 11), we have a = 385 and b = 2</p><p>∴ a + b = 385 + 2 = <strong>387</strong></p>
Correct Answer: B