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Convergence tests for not always positive series

There is

So in the rare cases that you can't use the alternating series test, use any of the tests for positive series for $ \sum \lvert a_n \rvert$. If the problem also asks about convergence and absolute convergence, always consider the sequence $ \sum \lvert a_n \rvert$ separately.

Which of the following series converge? Which of them converge absolutely?

  1. $ 1 - \frac 1{\sqrt{2}} + \frac 1{\sqrt{3}} - \dots$
  2. $ \sum_{k=0}^\infty \frac{(-1)^k k}{\sqrt{k^3 + 2}}$
  3. $ \sum_{n=0}^\infty (-1)^n\frac{n}{e^n}$
  4. $ 1 - \frac 12 + \frac 18 - \frac 1{16} + \frac 1{64} - \frac 1{128}
(The last one is a bit more tricky than the others.)

Arend Bayer 2006-12-15