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Let L = lim_{x->0} ( (1 - cos x)(1 - cos 2x)(1 - cos 3x) ) / x^6.…
Question
Let L = lim_{x->0} ( (1 - cos x)(1 - cos 2x)(1 - cos 3x) ) / x^6. Writing L = m/n in lowest terms with positive integers m, n, find m + n.
✓ Verified answer: 11checked by our engine — not a guess
Step-by-step solution
For each factor, 1 - cos(kx) = (k^2/2)x^2 + O(x^4). To leading order the product is ( (1/2)x^2 )·( (4/2)x^2 )·( (9/2)x^2 ) = (1/2)(2)(9/2) x^6 = (9/2) x^6.
Dividing by x^6 gives L = 9/2. Thus m + n = 9 + 2 = 11.
Final answer11
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