"Feel the bern" integral

Calculus Level 5

$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \int_{0}^{1} \dfrac{db \, de_{1} \, dr \, dn \, di \, de_{2} }{1- be_{1}rnie_{2} }$

If the above thing can be expressed in the form $\dfrac{a \pi^{b}}{c}$ , with $a, c$ as coprime positive integers, find $a+b+c$ .

Details and assumptions:

There's no $e \approx 2.71828...$ or $i = \sqrt{-1}$ . The $b, e_{1}, r, n, i, e_{2}$ are all variables.

The answer is 952.

1 solution

Otto Bretscher
Apr 29, 2016

Viva Bernie! Long live the Revolution!

This integral involving $n$ variables is $\zeta(n)$ ; see formula 10 here . In Bernie's case we have $\zeta(6)=\frac{\pi^6}{945}$ , so that the answer is $1+6+945=\boxed{952}$ .

We can derive this result by expanding the integrand into a geometric series: $\frac{1}{1-xy}=1+xy+x^2y^2+...$ in the case of $n=2$ , and then integrating term by term: $1+\frac{1}{2^2}+\frac{1}{3^2}+...=\zeta(2)$ . For larger $n$ it's "more of the same."

"Feel the bern" integral

The answer is 952.

1 solution

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