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Compute \(\frac{1}{\log_2(100!)} + \frac{1}{\log_3(100!)} + \frac{1}{\log_4(100!)} + \dots + \frac{1}{\log_{100}(100!)}.\)

 Apr 10, 2019
 #1
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sumfor(n, 2, 100, (log(n) / log(100!)) = 157.97 / 157.97 = 1

 Apr 10, 2019
edited by Guest  Apr 10, 2019
 #2
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\(\sum \limits_{k=2}^{100}\dfrac{1}{\log_k(100!)} = \\ \sum \limits_{k=2}^{100}\dfrac{\log(k)}{\log(100!)} = \\ \sum \limits_{k=2}^{100}\dfrac{\log(k)}{\sum \limits_{j=2}^{100}\log(j)} =\\ \dfrac{\sum \limits_{k=2}^{100} \log(k)}{\sum \limits_{j=2}^{100} \log(j)} = \\ 1\)

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 Apr 10, 2019

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