I Love My PhD, but I Am Tired: Burnout and Taking a Break

Feynman's trick turns this into an integral of digamma functions. A geometric series and a Gamma-function product reach the closed form ln 5 − (1/3) ln 2 without one, and the same steps show when a variant will or will not simplify.

Grimmett and Welsh's prime-faced die exercise has a two-line proof and a hidden assumption. Here is where the prime does its work, and where the claim breaks.

Four integrals, each worked both ways, and the chain-rule test that tells you when to reach for the larger chunk.