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Making it more compact, it seems to me that the equation you consider can write $$F(k)=\sum_{j=1}^m a_j^k-b=0$$ where all terms are known except $k$ (which I suppose to be a real number). I really do not know if any analytical method could apply to the problem. If I am right, then only numerical methods could be used (such as Newton as the simplest) ...


Stirling's Approximation says that $N!\approx\frac{1}{\sqrt{2\pi N}}\frac{N^N}{e^N}$, so $\alpha(N)\approx e/N$, and $\exp(-\alpha(N))\approx 1-e/N$, then $(1-\exp(-\alpha(N)))^N\approx e^N/N^N$


i have got $2-\left(1-e^{(N!)^{-1/N}}\right)^N$ after an ugly calculation.


Hint: Multiply the numerator and denominator by the factor $-e^{ikd \sin(\vartheta)/2}$.

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