5.6Absolute Convergence and the Ratio and Root Tests
2Absolute Convergence and the Ratio and Root TestsGiven any series an, we can consider the corresponding series|an| = |a1| + |a2| + |a3| + . . . whose terms are the absolute values of the terms of the original series.
3Absolute Convergence and the Ratio and Root TestsNotice that if anis a series with positive terms, then |an|= anand so absolute convergence is the same as convergence in this case.
4Example 1Show that the seriesis absolutely convergent.
5Example 2 Determine whether the series is absolutely convergent.121tan)1(nnnn
6Example 2-solution 12112112121211212112121convergentabsolutelyistan)1(convergentistan)1(test comparisonBy the12ptest with -pby theconvergentis112tan)1(1221tan1tan)1(nnnnnnnnnnnnnnnnnnbutnnnnnnnnn
7Example 3Show that the seriesis convergent, but it is not absolutely convergent.
8Absolute Convergence and the Ratio and Root TestsExample 2 shows that the alternating harmonic series is conditionally convergent. Thus it is possible for a series to be convergent but not absolutely convergent.