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I need guidance in finding a variable P for which $ \sum _{n=4}^{\infty }\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p $ converges, or proof that there doesn't exist such P variable.

I've tried the direct comparison test, by defining the sequence $ b_n=\:\left(\frac{n\ln \left(n\right)-n}{\ln \left(n!\right)}\right)^p $ and the sequence $ a_n=\left(1-\frac{1}{\ln \left(n\right)}\right)^p $.

From here I'm not sure how to prove that $\sum _{n=4}^{\infty }\:a_n$ diverges.

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    $\begingroup$ Have you tried replacing $\ln(n!)$ with the Stirling formula? $\endgroup$
    – Dieter Kadelka
    Commented Jun 15, 2020 at 13:08
  • $\begingroup$ Have you tried the divergence test on $a_n$? $\endgroup$
    – Gabe Conant
    Commented Jun 15, 2020 at 13:16
  • $\begingroup$ @DieterKadelka I've tried using the Stirling formula, but I didn't see where it can lead. $\endgroup$
    – Roee
    Commented Jun 15, 2020 at 13:28
  • $\begingroup$ @GabeConant I've tried using the direct comprasion test on $a_n$ with the sequence $ (1-(1/n))^p $ which I know diverges, but I'm not sure how to prove it. $\endgroup$
    – Roee
    Commented Jun 15, 2020 at 13:32
  • $\begingroup$ To show that $\Sigma (1-(1/n))^p = \infty$ assume that $p \in \mathbb{N}$ and use the $\zeta$-function. $\endgroup$
    – Dieter Kadelka
    Commented Jun 15, 2020 at 13:52

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