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\mathrm{E}(\mathrm{U})=\sum_{i=1}^{\infty} \pi_{i} \mathrm{U}\left(\mathrm{W}_{i}\right)=\frac{1}{2} \times \ln (2)+\frac{1}{4} \times \ln (4)+\ldots=\sum_{i=1}^{\infty} \frac{1}{2^{i}} \ln \left(2^{i},\right)