Calculus-Sec-9-1-Solution


  9.1 Sequences and Series                  by SGLee - HSKim -SWSun  

 

1-3. Find a formula for the general term  of the sequence, assuming that the pattern of the first few terms continues.

 

1. 

 

 

      .

      Therefore, .




2. 

 

 

      .

      Therefore, .




3. 

 

 

      Therefore, .




4-15. Determine whether the sequence converges of diverges. If it converges, find the limit.

 

 4. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-1-4.html 

 

 







 5. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-1-5.html 

 

 




 6. 

           http://matrix.skku.ac.kr/cal-lab/cal-10-1-6.html 

 

 







 7. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-1-7.html 

 

 







 8. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-1-8.html 

 

 







 9. 

            http://matrix.skku.ac.kr/cal-lab/cal-10-1-9.html 

 

 







 10. .

              http://matrix.skku.ac.kr/cal-lab/cal-10-1-10.html 

 

 







11. 

 

 

       Here,  and .

       Thus, .

       Hence, this series is divergent.







12. 

 

 

      

      Thus, 

      Hence, this series is divergent.







13. 

 

 

      

      Hence, this series is divergent.







14. 

 

 

      

      By L'Hopital's Rule,

      

      Hence, this series is convergent.







15. 

 

        

      Since ,

      Therefore, 

      Hence, this series is convergent.







16. Investigate the sequence  defined by the recurrence relation  for .

      In particular, show that  .

 

          

      Therefore, 




17. Let .

     (a) Determine whether  is convergent.

     (b) Determine whether  is convergent.

 

      (a)

         

        Therefore,  is convergent.

     (b)

        

       Therefore,  is divergent.










18-24. Determine whether the following series is convergent or divergent. Find the sum if it is convergent.

 

18. 

 

 

    

       Thus,  is divergent.




19. 

      http://matrix.skku.ac.kr/cal-lab/cal-10-1-19.html 


 

This is a geometric series and since , this series is convergent.

      And, .




20. 

 

 

       

      But,  is not convergent and  is convergent.

      Thus,  is divergent.




21. 

 

     .

      Therefore,  is convergent to .




22. 

 

      By L'Hopital's Rule,

        

       Therefore,  is divergent.




23. 

 

 

     Since , this geometric series with  () is convergent.

      And, .




24. 

 

       

      As 

      Thus,  is divergent.




25-26. Express the number as a ratio of integers.

 

25. 

 

      




26. 

     http://matrix.skku.ac.kr/cal-lab/cal-10-1-26.html

 

 

   




                                                      Back to Part I