Greetings,
I'm having trouble applying the tests for convergence on these series; I can never seem to wrap my head around how to determine if they're absolutely convergent, conditionally convergent or divergent.
a) $\displaystyle \sum_{k=1}^{\infty}\frac{\sqrt{k}}{e^{k^3}}$.
b) $\displaystyle \sum_{k=2}^{\infty}\frac{(-1)^k}{k(\ln k)(\ln\ln k)}$.
Answer
Hint: I imagine that the Ratio Test is part of your toolbox. Problem (a) should yield fairly easily to that.
For (b), note that the given series is (almost) an alternating series. (The first two terms have the same sign.) To prove that the series does not converge absolutely, a tool that works is the Integral Test. Try differentiating
$\ln(\ln(\ln x))$.
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