10.3 Alternating Series and Absolute Convergence

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


1. .

 Sol)

      By the Integral Test,

      is divergent.

     By Alternating Series Test, the series is divergent.


2.

 Sol) Note that iff ,

By Alternating Series Test, the series is divergent.


3. .

 Sol)

     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

 Sol) Note that

 

var('n')

u(n)=(-1)^(n+1)*n/(n+1)

limit(abs(u(n)), n=+oo)

 ind


5. .

 Sol) By the Integral Test,

      is divergent.

    By Alternating Series Test, the series is divergent.


6. .

 Sol)

     

     By the Ratio Test, this series is absolutely convergent.

     Hence, is convergent.


7. .

 Sol)

     By Alternating Series Test, the series is divergent.


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

 Sol) 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. .

 Sol)

     

     Hence, the series is divergent.


10. .

 

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

 Sol)   Divergent by the Ratio test.

 

var('n')

u(n)=exp(-n)*factorial(n)

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

  +Infinity


11. .

 Sol) .

     Since for all , we have .

     Thus

     Let

     

      is absolutely convergent.

     By the Comparison Test,

      is absolutely convergent, and hence convergent.


12. .

 Sol)

     By the root test,

     

      is absolutely convergent, and hence convergent.


13. .

 Sol) .

    

     We don't know that this is absolutely convergent.

     .

     Here, for all and

     By the Alternating Series Test,

      is conditionally convergent.


14. .

 Sol) Then

     

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

     Consider

     Here, for all and

     By the Alternating Series Test,

      is conditionally convergent.