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January 18th, 2019, 08:49 AM  #1 
Senior Member Joined: Dec 2015 From: iPhone Posts: 387 Thanks: 61  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 19th, 2019 at 12:21 AM. 
January 19th, 2019, 12:27 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,298 Thanks: 1971 
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?

January 19th, 2019, 02:01 AM  #3 
Senior Member Joined: Dec 2015 From: iPhone Posts: 387 Thanks: 61 
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 02:07 AM. 
January 19th, 2019, 05:56 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,298 Thanks: 1971 
$e^{2.71} >15$, but I think a lot of arithmetic is needed to prove that.


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