My Math 324 professor said we should try this for fun. Can someone explain how to come up with the answer? I thought maybe do a Taylor approximation since e^x is ∑ x^n / n!, but then I got something divergent.

Please contribute.

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- December 5th 2009, 11:15 PMJamin2112Which is greater: e^π or π^e?
My Math 324 professor said we should try this for fun. Can someone explain how to come up with the answer? I thought maybe do a Taylor approximation since e^x is ∑ x^n / n!, but then I got something divergent.

Please contribute. - December 5th 2009, 11:23 PMVonNemo19
- December 5th 2009, 11:28 PMRockHard
This is very interesting, do you have to provide some sort of proof?

- December 5th 2009, 11:33 PMBruno J.

What do you think? - December 5th 2009, 11:59 PMsimplependulum
the answer is is greater

let

we have , when

and

consider

we can see when ,

its second derivative gives

means has only one local min.

come back to

from the above calculations .

we can see

equality holds when (obviously )

ps . add a constraint - December 6th 2009, 12:31 PMKrizalid
consider the function which is strictly decreasing for

since we get and that yields and finally since is a strictly increasing function, then and we're done. - December 6th 2009, 09:27 PMJamin2112