An algebra problem by أحمد الحلاق

Algebra Level 4

$\frac {\log_{10}2}2 + \frac {\log_{10} 8}{10} + \frac {\log_{10} 32}{50} + \frac {\log_{10} 128} {250} + \cdots + \frac {\log_{10} 2^{2n-1}} {2\cdot 5^{n-1}} + \cdots =\, ?$

Give your answer to 3 decimal places.

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