In Problem 1, we asked:

Without using a calculator, work out which is larger, or .

First let us think about strictly monotone increasing functions. A function is strictly monotone increasing if if and only if . Now, is strictly monotone increasing, therefore (why?) so is its inverse, for . Now let’s go back to the problem.

Suppose we want to prove that , for . Since is strictly monotone increasing, is logically equivalent to , i.e. , which is the same as , which is the same as . But since is strictly monotone increasing, this is the same as . So what have we shown? That if and only if . This is good, as now we can use calculus.

Consider . Then by the chain rule (check!),

So the only critical point of is at . This clearly is a maximum, as and as (by l’Hôpital; check!) But then for any and so *in particular* , which as we have shown is equivalent to . They are pretty close! and .

(Brownie points to Michael Nolan, but no cigar.)

(MG)

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Nice general proof…but can I not just use the Maclaurin/Taylor series, e^x=1+x+x^2/2+…, so I know that e^x is greater than 1+x…

e^x>1+x

then let x=y-1 so

e^(y-1)>y

multiplying both sides by e gives

e^y>y*e

then let y=pi/e to get

e^(pi/e)>pi

now just raise both sides of the equation to the power e and we get

e^pi>pi^e