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Let L = lim_{n->infinity} n·( e - (1 + 1/n)^n ). The value can be…
Question
Let L = lim_{n->infinity} n·( e - (1 + 1/n)^n ). The value can be written L = (p/q)·e with p/q a positive fraction in lowest terms. Find p + q.
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Step-by-step solution
Let f(n) = (1+1/n)^n = exp( n·ln(1+1/n) ). With t = 1/n, n·ln(1+t) = (1/t)(t - t^2/2 + t^3/3 - ...) = 1 - t/2 + t^2/3 - ....
So (1+1/n)^n = e·exp(-t/2 + t^2/3 - ...) = e·(1 - t/2 + O(t^2)) = e - (e/2)/n + O(1/n^2).
Then e - (1+1/n)^n = (e/2)/n + O(1/n^2), and multiplying by n gives L = e/2.
Thus p/q = 1/2 and p + q = 3.
Final answer3
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