How can I determine the convergence/divergence of $\sum_{k=2}^{\infty} \frac {1}{\ln\left({k!}\right)}$?
Convergence of infinite series involving factorials
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$\begingroup$
calculus
sequences-and-series
convergence
factorial
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0Do you know Stirling's formula? – 2017-02-20
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0@carmichael561 No. Also, the only tests for convergence that I know are the Integral Test, Comparison Test, and Limit Comparison Test. – 2017-02-20
1 Answers
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Hint. $\ln(k!) \leq k \ln k.{}{}$
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0Thank you. Then I can simply use the Comparison Test to prove that the series diverges. However, how can I prove that ln(k!)≤k*lnk? – 2017-02-20
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0I can give you the answer, but you've only had five minutes to think about it. Are you sure you want to hear it? – 2017-02-20
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0No it's fine. I already figured it out. I wrote ln(k!)<=k*lnk. Then used log properties to write ln(k!)<=ln(k^k). After that I cancelled the logs to get k!<=k^k which is obviously true. – 2017-02-20
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0Okay, but your chain of implications should start with $k! \leq k^k$ and go in the other direction. – 2017-02-20