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**Soroban** Hello, PapaSmurf!

We have: .$\displaystyle S \;=\;\ln(\tfrac{1}{3}) + \ln(\tfrac{2}{4}) + \ln(\tfrac{3}{5}) + \ln(\tfrac{4}{6}) + \hdots + \ln(\tfrac{124}{126}) + \ln(\tfrac{125}{127}) + \ln(\tfrac{126}{128}) $

. . $\displaystyle S \;=\;\ln\left(\frac{1}{\rlap{/}3} \cdot \frac{2}{\rlap{/}4} \cdot \frac{\rlap{/}3}{\rlap{/}5} \cdot \frac{\rlap{/}4}{\rlap{/}6}\:\cdots\;\frac{\rlap{///}{124}}{\rlap{///}126} \cdot \frac{\rlap{///}125}{127} \cdot \frac{\rlap{///}126}{128}\right) $

. . . . $\displaystyle =\;\ln\left(\frac{2}{127\!\cdot\!128}\right) \;=\;\ln\left(\frac{1}{8128}\right) \;=\;\ln\left(8128^{-1}\right) $

. . . . $\displaystyle =\;-\ln8128 \;=\;-9.00307017$