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Conclude if the given series converges or diverges and justify your conclusion

by RoRi

Test the following series for convergence or divergence. Justify the decision.

    \[ \sum_{n=3}^{\infty} \frac{1}{ \log n \cdot (\log (\log n))^s}. \]

We look to apply the integral test. To that end let

    \[ f(x) = \frac{1}{x \log x (\log (\log x))^s}. \]

Then,

    \begin{align*} t_n &= \int_3^n \frac{1}{x \log x(\log(\log x))^s} \, dx \\ &= \begin{dcases} \frac{(\log(\log n))^{-s} (-1+ s + \log(\log n))}{-1+s} - \frac{(\log(\log 3))^{-s} (-1+s+\log(\log 3))}{-1+s} & \text{if } s \neq 1. \\ \log(\log(\log n)) - \log(\log(\log 3)) & \text{if } s = 1. \end{dcases} \end{align*}

Therefore, \{ t_n \} converges if s > 1 and diverges if s \leq 1. Hence,

    \[ \sum_{n=3}^{\infty} \frac{1}{n \log n (\log (\log n))^s} \]

converges if s > 1 and diverges if s \leq 1.

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