Without finding the numerical values, show which is greater, e^π or π^e.
The power series for e^x is 1 + x + x²/2! + x³/3! + ...
So we have e^x > 1 + x, for x > 0. (1)
Now set x = π/e - 1.
Then, by (1), e^(π/e - 1) > π/e.
Multiply by e: e^(π/e) > π.
Raise to the power e: e^π > π^e.