10.1 Sequences and Series

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


1. .

   Sol)

   .

   Therefore, .


2. .

   Sol)

   .

   Therefore, .


3. .

   Sol)

   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 

Sol)

 

var('x i n')

p1 = plot((1-x^3)/(2+3*x^2), (x, 1, 15), rgbcolor=(1,0,0))

p2 = list_plot([(1 - i^2)/(2 + 3*i^2) for i in range(0, 16, 1)], rgbcolor=(0,0,1))

show(p1+p2)


 

limit((1 - n^3)/(2 + 3*n^2), n=+oo)

  - Infinity


5. .

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



Sol)

 

var('x i n')

p1 = plot((x^2 - 2*x + 3)/(2*x^3 + 2), (x, 1, 50), rgbcolor=(1,0,0))

p2 = list_plot([(i^2 - 2*i + 3)/(2*i^3 + 2) for i in range(0, 51, 1)], rgbcolor=(0,0,1))

show(p1+p2)


   


 

limit((n^2 - 2*n + 3)/(2*n^3 + 2), n=+oo)

  0


6. .

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

Sol)

 

var('a b i n')

p1=plot(a^2/(3 + 2*a^2), (a, 1, 20), rgbcolor=(1,0,0))

p2=plot((-1)*b^2/(3 + 2*b^2), (b, 1, 20), rgbcolor=(1,0,0))

p3=list_plot([((-1))^i*i^2/(3 + 2*i^2) for i in range(0, 21, 1)], rgbcolor=(0,0,1))

show(p1+p2+p3)


   


 

limit(((-1))^n*n^2/(3 + 2*n^2), n=+oo)limit(ln(3*n)/ln(5*n), n=+oo)

  ind


7. .

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

 Sol)

 

var('x i n')

p1=plot((sin(x))^2/(3^x), (x, 1, 20), rgbcolor=(1,0,0))

p2=list_plot([(i,(sin(i))^2/(3^i)) for i in range(1,21,1)], rgbcolor=(0,0,1))

show(p1+p2)


    


 

limit((sin(n))^2/(3^n), n=+oo)

  0


8. .

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

 Sol)

 

var('x i n')

p1=plot(ln(3*(x))/ln(5*(x)), (x, 1, 40), rgbcolor=(1,0,0))

p2=list_plot([(i, ln(3*i) /ln(5*i)) for i in range(1,41,1)], rgbcolor=(0,0,1))

show(p1+p2)


 


 

limit(ln(3*n)/ln(5*n), n=+oo)

  1


9. .

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

 Sol)

 

var('x i n')

p1=plot((1- 1/x)^x, (x, 1, 20), rgbcolor=(1,0,0))

p2=list_plot([(i,(1- 1/i)^i) for i in range(1,21,1)], rgbcolor=(0,0,1))

show(p1+p2)


 


 

limit((1- 1/n)^n, n=+oo)

  e^(-1)


10. .

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

 Sol)

 

var('x i n')

p1=plot((3/x)^(2/x), (x, 1, 20), rgbcolor=(1,0,0))

p2=list_plot([(i,(3/i)^(2/i)) for i in range(1,21,1)], rgbcolor=(0,0,1))

show(p1+p2)


      


 

limit((3/n)^(2/n), n=+oo)

 1


11. .

   Sol)

   Here, and .

   Thus,  .

  Hence, this series is divergent.


12. .

   Sol)

    Thus,

  Hence, this series is divergent.


13. .

   Sol)

  Hence, this series is divergent. 


14. .

   Sol)

   By L'Hopital's Rule,

   

  Hence, this series is convergent.


15. .

   Sol)

   Since ,

   Therefore,

  Hence, this series is convergent.


16. Investigate the sequence defined by the recurrence relation for . In particular, show that .

   Sol)

   Therefore,


17. Let .

    (a) Determine whether is convergent.

    Sol)

        Therefore, is divergent.


    (b) Determine whether is convergent.

    Sol)

        Therefore, is divergent.


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


18. .

   Sol)

       Thus, is divergent.


19. .

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

   And, .


20. .

  Sol)

   But, is not convergent and is convergent.

   Thus, is divergent.


21. .

  Sol)

    Therefore, is convergent to


22.

   Sol) By L'Hopital's Rule,

   

   Therefore, is divergent.


23. .

   Sol) Since , this series is convergent.

   And, .

24. .

   Sol)

   As ,

   Thus, is divergent.


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


25.

   Sol)


26.

   Sol)