how can we find the sum of (ln n)/(2^n) for n=1,2,3... to infinity?

Question

We can find the near solution but can we prove it mathematically

Answer 1

∞
∑ln(n)/2^n ≈ 0.507833923
n=1
(adding 35 terms fixes 9 digits)

Answer 2

it is fast convergent, so do the first 20 and sum and you will be corect about 99.999%.