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Math Help - Factorial question

  1. #1
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    Factorial question

    How do you get \frac{1}{n!}\leq\frac{1}{2^{n-1}} for n>1?
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  2. #2
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    \displaystyle n \cdot (n-1) \cdot (n-2) \cdot \dots \cdot 3 \cdot 2 > 2 \cdot 2 \cdot 2 \cdot \dots \cdot 2 \cdot 2 (with \displaystyle n-1 terms on each side of the inequality)

    \displaystyle n! > 2^{n-1}

    \displaystyle \frac{1}{n!} < \frac{1}{2^{n-1}}.
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  3. #3
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    I can see the proof, but how do you get the first line ie what idea comes prior to that? why not 2.1, 2.4 or pi?

    nevermind. I got it.
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  4. #4
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    Fpr those who didn't "get it", both n! and 2^{n-1} have n- 1 factors (not counting the "1" in n!). Just compare the size of the factors. n! is the product of all numbers from n down to 2 while 2^{n-1} has all "2"s. 2= 2 while 3> 2 so 3(2)> 2(2), 4> 2 so 4(3)(2)> 2(2)(2), 5> 2 so 5(4)(3)(2)> 2(2)(2)(2), etc.
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