Is This a Valid Way of Finding Apery's Constant? Is this a valid way to find Apery's constant? Consider the power series generated by $\ln (1-x)$ Let $x= e^{ix}$. Integrate $f(e^{ix})$ two times. Then let $x=\pi$. By doing these manipulations do you get $\zeta(3)$ where $\zeta$ is the Riemann Zeta function?

If you integrate $\ln(1-e^{ix})$ twice, you get (up to a constant and an $x$ term) $\text{polylog}(3,e^{ix})$.  Evaluate this at $x = \pi$, and according to Maple the result is $-3\; \zeta(3)/4$.
