Sophomore’s Dream —The Peak of Mathematical Elegance

For this article, I take you back to the end of the 17th century — Johann Bernoulli’s time. Now, before I begin, I should present a word of caution. The Sophomore’s Dream is one of the most beautiful results in elementary mathematics and once you learn of it, there’s no going back. You will forever be lost in its beauty.

Anyway, now that the warning is out of the way, we can indulge in the pinnacle of elegance. What is the Sophomore’s Dream? Well, it’s a pair of identities that seem ‘too-good-to-be-true’ but are, in fact, true. Namely,

It’s special because the integral of the function is equal to its sum, only with different bounds. But to prove this will take some effort because this integral does not have an anti-derivative in terms of elementary functions.

Proof

Well, to begin. We represent the integrand in terms of the exponent function:

In order to reduce our problem, we turn to a Taylor series expansion of the integrand:

Then, since this series demonstrates uniform convergence, we can switch the order of integration and summation to yield:

The more experienced reader will see here that it is possible to reduce this integral significantly by means of a simple substitution and some elementary manipulations:

Now, if you notice, the integral on the RHS is simply the Gamma function with argument n+1. And, it is well known that Γ(x) = (x — 1)!. Substituting this result into the summation yields very simply:

With this, the proof is complete. The integral is equal to the sum and the Sophomore’s Dream is indeed true. Similarly, the second identity can be trivially reached following the same process. The methods are entirely analogous.

For the more interested reader, however, a second way to do this would be without turning to the Gamma function. Instead, by integrating by parts the summand of the Taylor series in the second step and making use of the falling factorial function, it is possible to reach the same result. However, a little bit of inductive reason is required. It was actually this method that Bernoulli used to prove these identities. The method in this article was discovered much later, after the world had been introduced to the Gamma function.

Now, unfortunately, this sum doesn’t have a closed-form representation. So, computing the sum is just an approximation. On the bright side though, the sum demonstrates some incredibly quick convergence and even summing just the first nine terms leaves an error of the order 10⁻¹⁰:

With this, the proof for the Sophomore’s dream is complete. You might be thinking that there should be a Freshman’s Dream. And you would be right. However, while it is similarly an elegant identity, it is simply not true. The interested reader can learn more here: Freshman’s Dream.

Thank you for reading and I hope you have a great day!