Calculus-Sec-9-3-Solutio


  9.3 Alternating Series and Absolute Convergence 

                                                            by SGLee - HSKim - SWSun

 

1-7. Test for convergence of the following alternating series:

 

1. 

 

 

       Since  is divergent, the series is divergent by the Integral Test.




2. 

 

 

      Note that  iff 

      

      By Alternating Series Test, the series is divergent.




3. 

 

 

       

      It satisfies  because  and .

      By the Alternating Series Test, the series is convergent.




 4. 

http://matrix.skku.ac.kr/cal-lab/cal-10-3-4.html 

 

      Note that 




5. 

 

 

       By the Integral Test,

       is divergent.

      By Alternating Series Test, the series is divergent.




6. 

 

      

      

      By the Ratio Test, this series is absolutely convergent.

      Hence,  is convergent.




7. 

 

 

       

      By Alternating Series Test, the series is divergent.




8. For what values of , the series  is convergent.

 

 

      By Exercise 5, when , this series diverges.

      Let  and .

      For  for all x.

      And for 

      Thus, by the Alternating Series Test, for , this series is convergent. If , this series is divergent.




9-14. Test whether the series is absolutely convergent, conditionally convergent, or divergent.

 

9. 

 

       

      

      Hence, the series is divergent.




 10. 

              http://matrix.skku.ac.kr/cal-lab/cal-10-3-10.html 

 

      Divergent by the Ratio test.




11. 

 

       .

      Since  for all , we have .

      Thus 

      Let 

      

       is absolutely convergent.

      By the Comparison Test,

       is absolutely convergent, and hence convergent.




12. 

 

      

      Using the root test,

      

       is absolutely convergent, and hence convergent.




13. 

 

       .

      

      We do not know if this series is absolutely convergent, when using the ratio test. Let us try another      test.

      .

      Here,  for all  and 

      By the Alternating Series Test,

       is conditionally convergent.




14. 

 

 

       Then

      

      This means we cannot conclude convergence of this series using the Ratio Test.

      Consider 

      Here,  for all  and 

      By the Alternating Series Test,

       is conditionally convergent.




                                                             

                                                               Massachusetts Institute of Technology

                                                              (MIT), Boston/Cambridge, MA, USA

 


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