To which series does it relate to?

Calculus Level 4

The value of the infinite series \[\sum _{ n=0 }^{ \infty } \frac{1}{{2^{n+1}}{(n+1)}} = \frac{1}{2}+\frac{1}{8}+\frac{1}{24}+\frac{1}{64}+\text{ }...\] is (give your answer with 4 significant digits):

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