CALCULUS 2
Name: _____________________________
WORKSHEET 11.1
Evaluate the sum:
5
1.
 3n  1
n 2
2
2.
3
4
n
3.
n  2
  1
n
n 1
n!
n 1
Rewrite the series using sigma notation:
4. 6  16  26  ...
5. 120  60  30  ... 
15
16
1 4 9 16
6.   

 ...
3 9 27 81
Rewrite the sequence using bracket notation:
7. 11 , 7 , 3 , ...
8. 4 , 6 , 9 , 13.5 , 20.25 , ...
27  
64 
 1  8 
9. 1  , 1  , 1  , 1  ,...
7 
9 
 3  5 
Determine if the sequence converges or diverges. If it converges, find the value:
 2n  3 
10. 

 3n  1 
 4n  3n 2 
11. 

3
 5n  1 
13. n 2 e 3n 
14. 0 ,
1 2 3
, ,
, ...
16 81 256
 n 
12. 

 ln n  
1
1
1
15. 1 , , 1 , , 1 , , ...
2
3
4
CALCULUS 2
Name: _____________________________
WORKSHEET 11.2
Using the appropriate test for monotonicity, define the sequence as increasing, decreasing, nondecreasing, non-increasing, eventually increasing or eventually decreasing. If the sequence eventually
increases or decreases, determine the value for which it changes:
Use the difference test:
 n 1
1. 

n  2
2.
2n
2

 3n
3.
n
2
 3n

Use the ratio test:
 n 1
4. 

n  2
 5n 
5.  n

 4 1
 n! 
6.  n 
7 
Use the derivative test:
 n 1
7. 

n  2
1

8. 3  
n

9.
  4n 
11. 

n  3
 n  5!
12. 

 n  1! 
Use any test:

10. n 3  3n 2

tan n
1
CALCULUS 2
Name: _____________________________
WORKSHEET 11.3
Evaluate the sum, if possible:

1.
1
 

n 0  7 

4.

k 1
n
 4 k 1 
 k 1 
7 

2.

k 1

5.
 3
2 
 4
e
 

n 1   
k

3.
n 1
6.
n9
  1  
n 1
8

 n 2
 1
 

4
n 1 
n
Rewrite the decimal as a fraction:
7. 0.2121212121….
8. 0.013013013013…
9. 6.222222…
Determine if the series converges or diverges by evaluating the sequence of partial sums:

10.
1 
1
 


k 1
k 2  k
n 5

n 1 n  2

11.
12.


k 2

3
  k  1
2

3
k2




CALCULUS 2
Name: _____________________________
WORKSHEET 11.4
Determine whether the series converges or diverges. Evaluate the sum, if possible:

1.
 n  2


k 1

2.
2
n 1
4.

1
3
 
4
k 1
k
10.
 4  3k 
3
k 1
2

2
n 1 1  n
6.
76
n
11.
 ke
n 1



12.

k 2
k 1
2
15.
2
k2
k 2 1

k 1 k  1

1
7
 4k  2

3k 2
 kln k 
k 2
1
n3
9.
k 2
14.
5
n7
n 0
n2 1
2
5
8.

13.

 n 1 2

n
2
3

n 1
n 1
n

3.

5.
7.
n 0


k2
1 k3
CALCULUS 2
Name: _____________________________
WORKSHEET 11.5
Determine whether the series converges or diverges. Evaluate the sum, if possible:

1.
6n

n 1 n!

4.

k 1
3k
k5
5n
7. 
n 1 3n!


2.

k 1

5.

n 1


n2

n 1 n!
k!
k 10
3.
n3
2n !
n2 1
6.  n
2
n 1
1
8.  k  
k 1  3 

k
9.
1 2 3
4
 

 ...
3 9 27 81
CALCULUS 2
CHAPTER 11 REFERENCE SHEET
SEQUENCES
Convergence or Divergence
evaluate lim
n 
same
same
small
big
big
small
(use L’Hopital’s Rule, if necessary)
converges to the ratio of the leading coefficients
converges to 0
diverges
Monotonicity
Difference Test
evaluate a n 1  a n
> 0 incr. ≥ 0 non-decr. < 0 decr. ≤ 0 non-incr.
Ratio Test
evaluate
a n 1
an
> 1 incr. ≥ 1 non-decr. < 1 decr. ≤ 1 non-incr
Derivative Test
evaluate f x 
> 0 incr. ≥ 0 non-decr. < 0 decr. ≤ 0 non-incr
.
SERIES
Convergence or Divergence
Partial Sums
study the sequence of the partial sums
if it converges, then the series converges
if it diverges, then the series diverges
Infinite Geometric Series
if −1 < r < 1, then the series converges to
a1
1 r
Divergence Test
an
evaluate nlim

≠ 0 the series diverges
= 0 inconclusive
Integral Test
let f x   a n and if f is positive and decreasing, then evaluate
if the integral converges, then the series converges
if the integral diverges, then the series diverges
P-series

of the form
1
k
k 1
p
if p > 1, then the series converges
if 0 < p ≤ 1, then the series diverges
Ratio Test
let   lim
n 
a n 1
an
if ρ < 1, then the series converges
if ρ > 1, then the series diverges
if ρ = 1, then the test is inconclusive

 f x dx
k
CALCULUS 2
CHAPTER 11 TEST REFERENCE SHEET
SEQUENCES
Monotonicity
Difference Test
> 0 incr. ≥ 0 non-decr. < 0 decr. ≤ 0 non-incr.
Ratio Test
> 1 incr. ≥ 1 non-decr. < 1 decr. ≤ 1 non-incr
Derivative Test
> 0 incr. ≥ 0 non-decr. < 0 decr. ≤ 0 non-incr
.
SERIES
Convergence or Divergence
Partial Sums
Infinite Geometric Series
Divergence Test
≠ 0 the series diverges
= 0 inconclusive
Integral Test
if the integral converges, then the series converges
if the integral diverges, then the series diverges
P-series
if p > 1, then the series converges
if 0 < p ≤ 1, then the series diverges
Ratio Test
if ρ < 1, then the series converges
if ρ > 1, then the series diverges
if ρ = 1, then the test is inconclusive
CALCULUS 2
Name: _____________________________
PRACTICE CHAPTER 11 TEST
Evaluate the sum:
5
2
1.
 2n
2.
n  3
n 1
  1
n 1
n2
n  1!
Rewrite the series using sigma notation:
3.  18  11  4  ...
4. 2 
3 4
5
10


 ... 
8 27 64
729
Rewrite the sequence using bracket notation:
5.
1 1 3
81
, , ,...,
6 2 2
2
6. 
1 2 6 24 120
, , , ,
...
4 1 0 1
4
Determine if the sequence converges or diverges. If it converges, find the value:
4 n  5
7. 

5
7  n 
8.
n e 
3
2 n
Using the appropriate test for monotonicity, define the sequence as increasing, decreasing, nondecreasing, non-increasing, eventually increasing or eventually decreasing. If the sequence eventually
increases or decreases, determine the value for which it changes:
Use the difference test:
Use the ratio test:
Use the derivative test:
n  3
9. 

 n 
3 
10.  
 3n!
en 
11.  
n
n
Use any test:


12. 3n 2  12n  1
 4n 
13.  n 1 
2 
Evaluate the sum, if possible:

14.   
n 0  3 



n
15.
n 1

n 4
  1  n 1 
5
n 1

Rewrite the decimal as a fraction:
16. −5.027027027027…
Use the appropriate test to determine if the series converges or diverges. Evaluate the sum, if possible:
Use the divergence test:
5n 2  3

2
n 1  2n  1
Use the integral test:

17.
Use the P-series test:

19.
 2n
3 4

18.
2
 n ln n 
n 2
Use the ratio test:
20.
n 1

3n !
n 1
n2

Use any test:

21.

n 1

23.
n2
en
1
3
n 1
n
3

22.
n3

n
n 1 10
24.


n 1
4
n5