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Problem : Give an example of a series that converges but does not converge absolutely.

Consider the series

1 - + - + ...    

Convergence follows from the alternating series test, whereas absolute convergence fails because the harmonic series diverges.

Problem : Prove that (- 1)ne-n2 converges.

The result follows from the alternating series test by noting that e-(n+1)2≤e-n2 and that e-n2 = 0.

Problem : Determine whether or not

(- 1)n    

converges.

The series converges by the alternating series test, since the absolute value of the n-th term in the series is

   

Notice that the convergence is not absolute.
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