Determine whether the series converges or diverges

4. Determine whether the series converges or diverges. For convergent series, find the sum
of the series.
8. Determine whether the series converges or diverges. For convergent series, find the sum
of the series.
10. Determine whether the series converges or diverges. For convergent series, find the sum
of the series.

20. Determine whether the series converges or diverges. For convergent series, find the sum
of the series.
28. Determine all values of c such that the series converges.
0
2
1
k
ck




C=k
Section 8.3
4. Determine convergence or divergence of the series.
(a)
8
4
2 4
k
k





4 2
2 4 1 2
1 1
1
1
1
lim lim
1 2
2
lim
lim(ln ]
ln ln1
R R
x x
R R
R
u
R
R
dx
u x
du dx
du
u
diverges
 
 




 

 
 

4
2 4
1
1
4 4
2 4 1 4
1
4 ( )
lim ( ) ( ) 1 0
2 4 ( )
...
k
k
k
k
k
k
a
k
k
b k
k
k
a
b
k
x
k
both diverge





    

(b)
2
6
4
(2 4 )
k
k




2
2
2 2
2
2
4
4 16
1
2
1
4 4
1 16
2
4 16 16 1
4 ( )
lim ( ) ( ) 1/ 4 0
4 16 16 ( )
...
k
k
k
k k
k
k
a
k k
b
k k
k
k
a
b
k
x
k k
both converge
 
 



    
 
10. Determine convergence or divergence of the series.

(b)
2
5
0
1
1
k
k
k





2
2 2
5/2 1/2
5
1/2
1 2 1/2 5/2 1/2 5/2
5 5 5/2 5/2
1 1
1
1
1
1 /
lim 1 0
1
1 1 / 1 1/
k
k
k
k
k k
a
k k
k
b
k
k k k k k
x
k k k k
diverges


  



 
   
  