If \( \sum_{r=0}^{n}(-1)^{r} \cdot{ }_{n}\left[\frac{1}{2^{r}}+\fra...

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If \( \sum_{r=0}^{n}(-1)^{r} \cdot{ }_{n}\left[\frac{1}{2^{r}}+\frac{3^{r}}{2^{2 r}}+\frac{7^{r}}{2^{3 r}}+\ldots . .\right. \).
to \( m \) terms \( ]=k\left(1-\frac{1}{2^{m n}}\right) \), then find the value of \( k \).
\( \mathrm{P} \)
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