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
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.