Power trip

Here’s a mathematical paper of which I can smugly say I understood every word.

Ha ha. No, seriously, I like it. A visual proof that e^\pi>\pi^e.

But as Math with Bad Drawings pointed out someone else pointed out, the proof doesn’t depend on \pi but can be generalized to any number > e, and as Math with Bad Drawings pointed out, a similar proof works for any positive number < e. That is, for all a > 0 and a\ne e, e^a>a^e.

Proof here if you don’t want to click through to MwBD:

 

If a > e,

\displaystyle \frac{1}{e}(a-e) = \frac{a}{e}-1 > \int_e^a \frac{1}{x}dx = \ln(a)-\ln(e) = \ln(a)-1

so

\displaystyle \frac{a}{e} > \ln(a)

or

\displaystyle a > \ln(a^e)

or

\displaystyle e^a > a^e

and similarly, if b < e,

\displaystyle \frac{1}{e}(e-b) = 1-\frac{b}{e} < \int_b^e \frac{1}{x}dx = \ln(e)-\ln(b) = 1-\ln(b)

so

\displaystyle \frac{b}{e} > \ln(b)

or

\displaystyle b > \ln(b^e)

or

\displaystyle e^b > b^e

QED

 

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