Infinite series for $\pi \mathrm{/}3$ and other identities

Using techniques from calculus, we combine classical identities for $\pi$, $\ln 2$, and harmonic numbers, to arrive at a nice infinite series formula for $\pi \mathrm{/}3$ that does not appear to be well known. In addition, we give twenty-seven related identities involving $\pi$ and other irrational numbers. Posted in celebration of Pi Day 2022.