Which one is greater?

 Imagine, it’s a life or death question. And you don’t even have a calculator. How do you even make sense of the given question? But mathematics has this way of surprising us, and often, it surprises us with connections we’d have never imagined. In this article, I’ll present to you two methods that you can use such connections to easily get to the answer to the above question.

 1.Taylor comes to rescue!

 Ever herad of Taylor expansion? Well, it’s nothing more than approximating non-polynomial functions using polynomials. Yes, all the sine, cosine, log, and exponential functions have a Taylor approximation of their own. More on that later! I think, not necessarily knowing the name for the expansion, everyone would be well aware of the expansion of e^x, the exponential function. You can even derive the expansion easily using the definition of e and binomial expansion. Why don’t you try it on your own before having a peek at the expansion below?

Now, as you can see, the expansion is an infinite series and the accuracy of our approximation increases as we increase the number of terms we take into consideration.

 Look closely at the expansion and perhaps, you’d not argue with me if I write down the following inequality:

The sum of the first two terms should always be less than the total sum that converges at infinity. It’s as simple as that. Now, let us do a really clever selection of the value of x in the inequality so as to suit our purpose, shall we?

Here it is, with just a keen eye at the expansion of e^x and a clever selection of x, we easily found the solution to a question that just saved your life. Interesting, eh?

2. Calculus comes to rescue

How about re-writing the original question so as to see it through the lens of calculus?

Then, since the powers of both terms are equal, the comparison further reduces to:

Did you just have butterflies in your stomach? Because I had! We just converted the problem into a problem of looking at a function y =x^(1/x) and deciding whether it’s greater at x = e or x = π.

So, draw a graph and tally the values? Well, no! Let’s just say you don’t have access to Desmos at the moment. Life is full of constraints, all right! 

Then, amidst all this chaos of constraints and frustrations, you get an idea. Let’s calculate the maxima, you tell me. Of course, why didn’t we think of it earlier, I say. So goes the following series of lines using basic differentiating(firstly operating ln on both sides and using product rule) and setting dy/dx = 0 to get the maxima. You even calculate the second derivative to see if it’s actually a maxima and not a minima.

Then to calculate the maximum, you set dy/dx = 0,

Alright, the result baffles you! Maximum at x = e means that the e^e is the maximum value for the function, thus it is clear that:

So, a neat idea and precise use of calculus led us straight to the answer, didn’t it? So that’s it for today’s article. Perhaps, next time a question like this is asked to you in a life/death situation, you’ll quickly make necessary connections and solve it. 

Everything’s interesting, isn't it?

-Manoj Dhakal