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Homework Statement
Let ##x_n## be the solution to the equation
##\left( 1+\frac{1}{n} \right)^{n+x} = e##
Calculate ##\lim_{n\to \infty} x_n##
Homework Equations
N/A
The Attempt at a Solution
Since ##\lim_{n \to \infty} \left(1+ \frac{1}{n} \right) = e## that tells me that ##\lim_{n\to \infty} x_n = 0## but the answer in the book says it should be ##\frac{1}{2}## which I don't understand at all.
This is also in a section about Taylorexpansion which suggest I should use that somehow. If I expand on x i get something like this
##\left( 1+\frac{1}{n} \right)^{n+x} = \left( 1+\frac{1}{n} \right)^{n}\cdot \left( 1+\frac{1}{n} \right)^{x} =\left( 1+\frac{1}{n} \right)^{n}\cdot (1+\ln (1+\frac{1}{n})x + O(x^2) )##
which doesn't seen to help me and I still don't see why ##x_n\to \frac{1}{2}##.
Some advice? :)
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