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 + ... to reduce the numerator, then factor out x² to find the limit. The sum ∑cos(kx) telescopes to a specific form when expanded.
<p><strong>Step 1:</strong> Expand each cosine 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 from k=1 to 10:</p><p>∑(k=1 to 10) cos(kx) = 10 - (x²/2)∑(k=1 to 10)k² + O(x⁴)</p><p><strong>Step 3:</strong> Calculate ∑(k=1 to 10)k²:</p><p>∑k² = 10(11)(21)/6 = 385</p><p><strong>Step 4:</strong> Substitute into the numerator:</p><p>10 - ∑cos(kx) = 10 - [10 - (x²/2)·385 + 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) = gcd(5·7·11, 2) = 1, we have a = 385 and b = 2</p><p>∴ a + b = 385 + 2 = <strong>387</strong></p>
Correct Answer: B