Maths Puzzle: What is greater? e^pi or pi^e!

Find out which one is greater,
e^π  or π^e  

condition is not to use calculator or make any calculation in powers of 2.73... or 3.14...

This is fairly simple mathematical riddle that needs understanding of high school level Mathematics. A friend of mine asked this to me[he was asked this in some interview], along with proper reasoning. And I realised, this fairly simple riddle becomes quite difficult for 'mathematics layman' like me! Thanks that I was not facing any such interview.

After some struggle, I managed to find the solution. Then googled to check what solutions are available for this on net. There were many solutions on the net ranging from 3/4 lines to a verbose description, using variety of methods (including the one I used) with most popular being use of derivatives.

The method I followed is as below:

Let π = e^k        ----------------- (I)

L H S	Comparison/Operation	R H S
eπ		πe
ee^k	Put π = ek	ek.e
ek		k.e
ek-1	Dividing both sides by e	k

 

Now, let’s go by definition of e, e = (1+1/x)^x , Where x is infinitely large, for the closest value, x can be considered as the huge  number.

Also, e^k =      (1+k/x)^x ------------------- (II)

Result in (II) is arrived from the fact that, as per definition, e is also equal to

(1+k/x)^x/k  for x/k which is infinitely large. Just, raising power by k to the both sides gives us equation in (II).

Applying (II) for (k-1) and Expanding (II) with binomial theorem,

e^k-1 =      (1+(k-1)/x)^x

       =     1+  x. (k-1)/x    + ^xC₂ . (k-1)²/2! + …. 

       =    1 + k – 1 + other terms involving (k-1)

        =   k + other terms involving (k-1)  --------------- (III)

Now, it is clear that for all positive values greater than 1, RHS of (III) is always greater than k. for k=1, other terms will be zero and it will be equal to k.

Hence, e^k-1 > k,  for all k>1.

As π > e, it is clear that k > 1.

Hence table completes as,

L H S	Comparison/Operation	R H S
eπ		πe
ee^k	Put π = ek	ek.e
ek		k.e
ek-1	Dividing both sides by e	k
ek-1	>	k
LHS	>	RHS
eπ	>	πe

 

Thus,

e^π  > π^e

- Chaitanya D Sangwai (09th July 2016).