The fifth power of Pi: new series representation involving the golden ratio and an application in physics

Although many series exist for \(\pi\) and \(\pi^2\), very few are known for higher (and in particular odd) powers of \(\pi\). In a previous work, we presented series representations for \(\pi^3\), in which the golden ratio appears in the prefactor. In this article, we derive, using a trigonometric series obtained by Euler, two representations of \(\pi^5\) involving infinite sums and the golden ratio. The methodology can be generalized in order to obtain further series, relating by the way \(\pi^5\) to other mathematical constants. We also mention an intriguing relation, published in 1951 in the journal ``Physical Review'' (and probably the shortest paper ever published in the journal), pointing out the fact that the ratio of proton to electron masses seems to be very close to \(6\pi^5\).