This is a homework question and I am not really sure where to go with it. I have a lot of trouble with sequences and series, can I get a tip or push in the right direction?
Prove that $(1 - \frac{1}{n})^{-n}$ converges to $e$
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sequences-and-series
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0@Ilya: deleted more like. I guess that my comment should have been to that answer. We should leapfrog deletes :-) – 2012-03-07
1 Answers
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You have: $ x_n:=\left(1-\frac1n\right)^{-n} = \left(\frac{n-1}n\right)^{-n} = \left(\frac{n}{n-1}\right)^{n} $ $ = \left(1+\frac{1}{n-1}\right)^{n} = \left(1+\frac{1}{n-1}\right)^{n-1}\cdot \left(1+\frac{1}{n-1}\right) = a_n\cdot b_n. $ Since $a_n\to \mathrm e$ and $b_n\to 1$ you obtain what you need.
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15Nice solution, uses almost nothing other than the definition of $e$. – 2011-11-14