JEEOlympiad
Let L = lim_{x->0} ( (1 + x + x^2)^(1/x) - e·(1 + x/2) ) / ( e·x^2 ).…
Question
Let L = lim_{x->0} ( (1 + x + x^2)^(1/x) - e·(1 + x/2) ) / ( e·x^2 ). Writing L = m/n in lowest terms (m an integer, n a positive integer, gcd=1), find |m| + n.
✓ Verified answer: 37checked by our engine — not a guess
Step-by-step solution
Let g(x) = (1/x)·ln(1+x+x^2). With u = x + x^2, ln(1+u) = u - u^2/2 + u^3/3 - .... Computing to order x^3: ln(1+x+x^2) = (x+x^2) - (x+x^2)^2/2 + (x+x^2)^3/3 - ... = x + x^2 - (x^2 + 2x^3)/2 + x^3/3 + O(x^4) = x + x^2/2 - (2/3)x^3 + O(x^4).
So g(x) = 1 + x/2 - (2/3)x^2 + O(x^3), and (1+x+x^2)^(1/x) = e·exp( x/2 - (2/3)x^2 + ... ). Let v = x/2 - (2/3)x^2. Then exp(v) = 1 + v + v^2/2 + O(x^3) = 1 + x/2 + ( -2/3 + 1/8 )x^2 + O(x^3) = 1 + x/2 - (13/24)x^2 + O(x^3).
Thus (1+x+x^2)^(1/x) - e(1 + x/2) = e·( -(13/24)x^2 + O(x^3) ), and dividing by e x^2 gives L = -13/24.
So m = -13, n = 24, |m| + n = 37.
Final answer37
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