10.2 Tests for convergence of series with positive terms

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


1. .

 Sol) 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


 Sol)

 

var('x')

f=x*exp(-x)

integral(f, x, 1, oo)

  2*e^(-1)   (converge to )

  Thus, the series converges.


3. .

 Sol) The function is continuous, positive on , and also decreasing. hence by Integral test, since for , so we can use the Integral Test.

      .

      Hence by Integral Test diverges.


4. .

 Sol) The function is continuous, positive on , and also decreasing, since   if , so we can use the Integral Test.

      .

      Hence by the Integral Test, converges.


5.

 Sol)

      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.


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


6. .

 Sol) 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. .

 Sol) We have already shown (in Exercises 3) 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.

     

     Thus the series converges for .


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

 Sol) 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. .

 Sol) , so the series converges by comparison with the -series


10. .

 Sol) Use the Limit Comparison Test with and .

     

     Since converges,  converges.


11. .

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

 Sol)

 

var('n')

u(n)=2^n*factorial(n)/n^n

limit(u(n+1)/u(n), n=+oo)

  2*e^(-1)


 

bool(2*e^(-1)<1)

  True


12. .

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

 Sol)

 

var('n')

u(n)=n^2/3^n

limit(u(n+1)/u(n), n=+oo)

  1/3


13. .

 Sol) 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?

 Sol) Use the Comparison Test.

     If for all , then is convergent.

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


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

 Sol) 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.