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January 18th, 2019, 07:49 AM   #1
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Integer part

How can I find the integer part of $\displaystyle e^e \;$? (without calculator)
e - Euler constant
What about approximation?

First, I tried one example for $\displaystyle 2e$ and $\displaystyle e^2$, to see whether it is possible.
$\displaystyle 3>e>\frac{5}{2}\;$ ,then multiply by 2
6>2e>5 so integer part of 2e is 5.

Last edited by skipjack; January 18th, 2019 at 11:21 PM.
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January 18th, 2019, 11:27 PM   #2
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How much basic arithmetic is acceptable, given that use of a calculator isn't allowed? Can you assume that e = 2.718... or does that have to be proved as well?
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January 19th, 2019, 01:01 AM   #3
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By calculator ,integer part of $\displaystyle e^e$ is 15.
But integer part of $\displaystyle {2.7}^{2.7}$ is 14 , same for 2.71 .

Last edited by idontknow; January 19th, 2019 at 01:07 AM.
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January 19th, 2019, 04:56 AM   #4
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$e^{2.71} >15$, but I think a lot of arithmetic is needed to prove that.
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