$$\sum_{k=0}^\infty {3^{k\ln k} \over {k^k}}$$
I need to determine the convergence of this series. I know it diverges, but how do I prove this?
2026-03-29 18:32:51.1774809171
Determine the convergence of the following series
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Use the Root test to get $$\lim_{k\to+\infty}(\frac{3^{k\ln k}}{k^k})^{1/k}=\lim_{k\to+\infty}\frac{3^{\ln k}}{k}=\lim_{k\to+\infty}(\frac{3}{e})^{\ln k}=+\infty>1$$ so on ... If you change $3$ with $2$ then it is convergent; interesting!