THE UNIVERSITY OF ZAMBIA
DEPARTMENT OF MATHEMATICS AND STATISTICS
MAT 2110-ENGINEERING MATHEMATICS I
Tutorial Sheet 5
1. Write the nth term for the sequences below:
1 1
1
(a) 1,  , ,  ,...
4 9 16
1 1 1
(b) 1, , , ,...
2 6 24
1 1 1
(c) 1,  , ,  ,...
3 5 7
3 7 15
(d) 1, , , ,... .
2 4 8
2. Determine whether the sequences below converge or diverge. If it converges, find
the limit.
n4  2
(a) an 
2 n
(b) an 
sin n
ln n
(c) an 
n
n
(d) an 
e2n
4n
3. Which of the sequences below are non-decreasing, non-increasing, bounded from
below, bounded from above, converge or diverge?
2n  1
(2n  3)!
n 1
(a) an  n (b) an 
(c) an 
n
2
(n  1)!
4. Identify the series which converge from the following series. Find the formula for
the nth partial sum and use it to find its sum.
2 2 2
2
(a) 2   
 ...  n1  ...
3 9 27
3
2 3 4
n
(b) 2     ... 
 ...
3 4 5
n 1
1
1
1
1
1



 ... 
 ... .
(c)
2.3 3.4 4.5 5.6
( n  1)( n  2)
5. Determine whether the following series converge:

3n
(a)  n 1 (b)
n0 5
 2 n 5n 

   (c)
4
n 0  3
(1) n
(d)

n
n 1 e




n 1
 1 
 5
n 
6. Find the sum of the following series:
1 
 1



 (b)
n
n 1 
n 1 

(a)
2n  1

2
2
n 1 n ( n  1)
ln(n  1)  ln(n  2)
n 1 ln( n  2).ln( n  1)


(c)

n 2  5n  2
.

2
2
n 1 n ( n  1)

(d)
7. Use the direct comparison test to determine the convergence or divergence of the
following series:




3n
ln n
1
1
1
(b)
(c)
(d)
(e)
.





3
2
n
n 1
n 1 2 n  1
n 1 2 n  1
n2
n 1 3n  2
n 1 2  1
8. Use the limit comparison test to determine the convergence or divergence of the
following series:

(a)
2n  1
(d)  n

n 1
n 1 5  1
n2  1
9. Use the ratio test to determine the convergence or divergence of the following
series:


5
(a)  sin( ) (b)  n
(c)
n 1
n 1 4  1

1
n

n!
(a)  n (b)
n 1 3
(1) n 1 n
(c)

2
n 1 n  1


1

6
n 

n 1  6 
n
(d)

(1) n 1  n3 
n 1
n2

n
10. Use the nth root test to determine the convergence or divergence of the following
series:

 2n 
(a)  

n 1  n  1 
n

(b)
e
n 0
3 n

(c)
n
 (ln n)
n2
n
11. Verify that the Integral test can be applied to the series below and use it to
determine convergence or divergence.





2
1
ln n
1
n
(a) 
(b)  3 (c) 
(d)  4
(e)  ne  n
n2
n 1
n2 n
n  2 n ln n
n 1 n  1
n 1
12. Find the radius of convergence and interval of convergence of the following
series:
( x  3) n 1

n 1
n  0 ( n  1).4
x n1

n
n 1 n.3


(a)
(b)
 n !( x  a)n
n 1
n( x  1) n

n
n 1 2 (3n  1)


(c)
(d)

(e)
(2n)! x 2 n

n!
n 1
13. Find the Maclaurin series and Taylor series at x  c up to the term with x 4 for
the following functions:
(a) f ( x)  sec x, c  
x
(b) f ( x) 
, c2
x 1
(c) f ( x)  e x x , c  1.
2
14. Use substitution, addition, Multiplication, differentiation or integration to find
Maclaurin series for the following functions:
x2
(a) f ( x)   1  cos x
2
(b) f ( x)  cos x
(c) f ( x)  e x  x
2
(d) f ( x)  sec 2 x
(e) f ( x)  sin x
15. Given that
2 n 1
x3 x5 x 7
n 1 x
arctan x  x     ...  (1)
 ..., | x |  1,
3 5 7
2n  1
evaluate the integral

1 arctan ( x 2 )
x
0
dx,
giving your answer such that the error is positive and does not exceed 0.01.
16. Use the Maclaurin series for ln |1  x | to approximate ln  32  so that the absolute
value error is less that 0.0001.