Compute \(\frac{1}{\log_2(100!)} + \frac{1}{\log_3(100!)} + \frac{1}{\log_4(100!)} + \dots + \frac{1}{\log_{100}(100!)}.\)
sumfor(n, 2, 100, (log(n) / log(100!)) = 157.97 / 157.97 = 1
+6251 \(\sum \limits_{k=2}^{100}\dfrac{1}{\log_k(100!)} = \\ \sum \limits_{k=2}^{100}\dfrac{\log(k)}{\log(100!)} = \\ \sum \limits_{k=2}^{100}\dfrac{\log(k)}{\sum \limits_{j=2}^{100}\log(j)} =\\ \dfrac{\sum \limits_{k=2}^{100} \log(k)}{\sum \limits_{j=2}^{100} \log(j)} = \\ 1\)
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