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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- Dec 5th 2009, 10:15 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. - Dec 5th 2009, 10:23 PMVonNemo19
- Dec 5th 2009, 10:28 PMRockHard
This is very interesting, do you have to provide some sort of proof?

- Dec 5th 2009, 10:33 PMBruno J.

What do you think? - Dec 5th 2009, 10:59 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 - Dec 6th 2009, 11:31 AMKrizalid
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. - Dec 6th 2009, 08:27 PMJamin2112