MATHS BITE: e^π > π^e

Consider the Taylor Series for Screen Shot 2017-02-25 at 9.32.27 PM.png:

Screen Shot 2017-02-25 at 9.32.23 PM.png

 

If x>0, then

Screen Shot 2017-02-25 at 9.36.03 PM.png

Letting x = π/e – 1 gives us

Screen Shot 2017-02-25 at 9.35.30 PM.png

Multiplying by e, we find that

Screen Shot 2017-02-25 at 9.37.10 PM.png

Raising everything to the power of e gives us the final, desired result:

Screen Shot 2017-02-25 at 9.38.11 PM.png

M x

 

 

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