Calculus-Sec-9-2-Solution


  9.2  Tests for Convergence of Series         by SGLee - HSKim-SWSun

 

1-5. Determine whether the series is convergent or divergent using the Integral Test.

 

1. 

 

 

The function  is continuous, positive, and decreasing on . The Integral Test applies.

      .

      Hence by the Integral Test,   is diverges.




 2. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-2-2.html 

 

 




 Thus, the series converges.

3. 

 

      

       is continuous and positive on ,

      and also decreasing since  for .

      Hence we can use the Integral Test.

     .

      Thus,  converges. And by the Comparison Test,  is also convergent.







4. 

 

 

     The function  is continuous, positive on , and also decreasing.

      Then, by Integral test, since  for , so we can use the Integral Test.

      .

       Hence by Integral Test  diverges.




5. 

 

 

       The function  is continuous, positive on ,

       and also decreasing, since   if  ,

       so we can use the Integral Test and .

       .

       Hence by the Integral Test,  converges.




6-7. Find the values of  for which the series is convergent.

 

6. 

 

 

       When  is continuous and positive on ,

       and also decreasing since  for  (  for ).

       Hence we can use the Integral Test.

       

      Thus, when  diverges.

      When  is continuous and positive on ,

      and also decreasing since  for ,

      so we can use the Integral Test,

       

      This limit exists whenever , so the series converges for .




7. 

 

 

     We have already shown (in Exercise 4) that when , the given series is divergent.

     Let us assume .

     The function  is continuous, positive on ,

     and also decreasing since  if ,

     so we can use the Integral Test and .

     

     Thus the series converges for .




8. Find all positive values of  for which the series  converges.

 

      The function  is continuous, positive on ,

      and also decreasing since  if  , so we can use the Integral Test:

      

      Thus   for .  The series converges for 




9-13. Test for convergence or divergence of the series.

 

9. 

 

 

  , so the series  converges by comparison with the -series .




10. 

 

 

      Use the Limit Comparison Test with  and .

      

      Since  converges,   converges.




 11. 

              http://matrix.skku.ac.kr/cal-lab/cal-10-2-11.html 

 

 




 12. 

              http://matrix.skku.ac.kr/cal-lab/cal-10-2-12.html 

 

 




13. 

 

 

      Use the Limit Comparison Test with  and .

      Then  and  are series with positive terms and

.

       Since  converges, converges.




14. If  is a convergent series with non-negative terms, is it true that  is also convergent?

 

 

      Use the Comparison Test.

      If  for all , then  is convergent.

      But, if  for all  then we cannot decide the convergence of .




15. If  is a convergent series with positive terms, is it true that  is also convergent?

 

      Yes. Since  is a convergent series with positive terms, ,

      and  is a series with positive terms (for large enough ).

      We have .

      Thus,  is also convergent by the Limit Comparison Test.

16. Find the radius of convergence of .

     http://matrix.skku.ac.kr/cal-lab/cal-10-2-16.html

 

 




17. Find the radius of convergence of .

      http://matrix.skku.ac.kr/cal-lab/cal-10-2-17.html

 

 




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