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       #Post#: 123--------------------------------------------------
       e IS IRRATIONAL
       By: eba95 Date: July 30, 2010, 6:59 am
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       e is irrational
       If e were rational, then e = n/m for
       some integers m, n. So then 1/e = m/
       n. But the series expansion for 1/e is
       1/e = 1 - 1/1! + 1/2! - 1/3! + ...
       Call the first n terms of this
       alternating series S(n). How good is
       this approximation to e? Well, the
       error is bounded by the next term of
       the alternating series:
       0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n
       +1)!
       But multiplying through by n!, you
       will see that
       0 < | integer - integer | < 1/(n+1) < 1.
       But there is no integer strictly
       between 0 and 1, so this is a
       contradiction; e must be irrational.
       Presentation Suggestions:
       Use the series expansion for 1/e as a
       fun fact on a previous day.
       The Math Behind the Fact:
       Anytime you have an alternating
       series in which the terms decrease,
       then each partial sum is not farther
       from the limit than the next term in
       the series!
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