Who Let The Logarithms In?

Calculus Level 5

ln423+ln633+ln843+ln1053+\dfrac{\ln 4}{2^{3}} + \dfrac{\ln 6}{3^{3}} + \dfrac{\ln 8}{4^{3}} + \dfrac{\ln 10}{5^{3}} + \cdots

You may use the approximations:
ζ(3)1.2020569\zeta(3)\approx 1.2020569 and ζ(3)0.198126 \zeta'(3)\approx-0.198126.

Notation: ζ()\zeta(\cdot) denotes the Riemann zeta function.

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