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Let L = lim_{x->0} ( (1+x)^(1/x) - e + (e x)/2 ) / x^2. The value of…
Question
Let L = lim_{x->0} ( (1+x)^(1/x) - e + (e x)/2 ) / x^2. The value of L can be written as L = (m/n)·e where m/n is a positive fraction in lowest terms. Find m + n.
✓ Verified answer: 35checked by our engine — not a guess
Step-by-step solution
Write (1+x)^(1/x) = exp( ln(1+x)/x ). Using ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + ..., we get ln(1+x)/x = 1 - x/2 + x^2/3 - x^3/4 + ...
So (1+x)^(1/x) = e · exp( -x/2 + x^2/3 - ... ). Let u = -x/2 + x^2/3 - .... Then exp(u) = 1 + u + u^2/2 + ... = 1 + (-x/2 + x^2/3) + (1/2)(x^2/4) + O(x^3) = 1 - x/2 + (1/3 + 1/8)x^2 + O(x^3) = 1 - x/2 + (11/24)x^2 + ....
Hence (1+x)^(1/x) = e - (e/2)x + (11e/24)x^2 + ....
Therefore the numerator (1+x)^(1/x) - e + (e x)/2 = (11e/24)x^2 + O(x^3), and dividing by x^2 gives L = 11e/24.
Thus m/n = 11/24, and m + n = 35.
Final answer35
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