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Let L = lim_{x->0} ( ln(1+x)·ln(1-x) + x^2 ) / x^4. Writing L = m/n…
Question
Let L = lim_{x->0} ( ln(1+x)·ln(1-x) + x^2 ) / x^4. Writing L = m/n in lowest terms (m an integer, n a positive integer), find |m| + n.
✓ Verified answer: 17checked by our engine — not a guess
Step-by-step solution
Use ln(1+x) = x - x^2/2 + x^3/3 - ...
and ln(1-x) = -x - x^2/2 - x^3/3 - ....
Their product up to x^4: the x^2 term is (x)(-x) = -x^2; the x^3 terms cancel by symmetry; the x^4 term comes from (x)(-x^3/3) + (-x^2/2)(-x^2/2) + (x^3/3)(-x) = -x^4/3 + x^4/4 - x^4/3 = (-1/3 + 1/4 - 1/3)x^4 = -(5/12)x^4.
So ln(1+x)ln(1-x) = -x^2 - (5/12)x^4 + O(x^6). Adding x^2 leaves -(5/12)x^4 + O(x^6); dividing by x^4 gives L = -5/12.
Thus m = -5, n = 12, |m| + n = 17.
Final answer17
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