# Determine the values of $p$ for which $\sum_{k=3}^\infty\frac1{k\ln k[\ln(\ln k)]^p}$ converges. [duplicate]

Determine the values of $$p$$ for which the given series converges: $$\sum_{k=3}^\infty\frac1{k\ln k[\ln(\ln k)]^p}.$$

• what convergence test is best here – Integrand Sep 16 at 16:33
• Use the integral test. – alex.jordan Sep 16 at 16:36

My answer may or may not be accurate. It's been a while since I've done power series stuff. I would recommend starting with the integral test because notice that the derivative of $$\ln(\ln k)$$ is $$\frac{1}{k\ln(k)}$$. So set up the integral as $$\int_{k=3}^\infty\frac1{k\ln k[\ln(\ln k)]^p}dk$$ Then perform a u substitution: $$u=\ln(\ln k)$$ and $$du=\frac{dk}{k\ln(k)}$$ so we have $$\int_{u=\ln(\ln(3))}^\infty\frac{1}{u^p}du$$ This is basically equivalent to a p series, which only converges if p is greater than 1. Perhaps this is how one is supposed to find the answer? Again, I am a bit rusty, and I'm sorry if this is not accurate.
• Integrale should go from $\ln\ln3$ to $+\infty$ – enzotib Sep 16 at 17:15