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\( \sum_{r=0}^{n}(-1)^{r} \cdot{ }^{n} C_{r} \frac{1+r \cdot \log _{e} 10}{\left(1+\log _{e} 10^{n}\right)^{r}}= \)

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\( \sum_{r=0}^{n}(-1)^{r} \cdot{ }^{n} C_{r} \frac{1+r \cdot \log _{e} 10}{\left(1+\log _{e} 10^{n}\right)^{r}}= \)
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