THE UNIVERSITY OF ZAMBIA
SCHOOL OF NATURAL SCIENCES
DEPARTMENT OF MATHEMATICS & STATISTICS
MAT 2110–Engineering Mathematics I
Tutorial Sheet 4
July 2024
1. Determine the convergence or the divergence of the following sequences:
o
n 1
o
n no
n
n
1−(−1)
n
nπ
√
(b) {an } = sin 2
(c) {an } = n 2 − 1
(d) {an } = (−4)
(a) {an } =
n!
n
2. Find the nth partial sum of each of the following series. Hence determine whether the series
converge or diverge:
4 4
4
4
+ +
+ · · · + n−1 + . . .
3 9 27
3
1
1
1
1
(b)
+
+
+ ··· +
+ ...
2.3 3.4 4.5
(n + 1)(n + 2)
2
2
2
2
+
+
+ ··· +
+ ...
(c)
2.4 3.5 4.6
(n + 1)(n + 3
(a) 4 +
3. Determine the convergence or the divergence of the following series:
∞
∞
∞
X
X
X
2n
1
√
(a)
(b)
(c)
n−0.5
n−1
3
3
n
n=1
n=1
n=1
(d)
∞
X
(−1)n
n+1
n=0
(e)
∞
X
en
3
n=1
(f)
∞
X
(−1)n n
n=1
2n + 1
4. Verify that the Integral Test can be applied to the following series and use it to determine
convergence or divergence:
∞
∞
∞
X
X
X
arctan n
n
1
(b)
(c)
(a)
2
2
4
n(ln n)
n +1
n +1
n=2
n=1
n=1
5. Use the Direct Comparison Test to determine the convergence or the divergence of the
following series:
∞
∞
∞
X
X
X
ln n
1
ln n
(a)
(b)
(c)
3
n
2n − 3
ln(ln n)
n=2
n=1
n=1
6. Use the Limit Comparison Test to determine the convergence or the divergence of the following series:
∞
∞
∞
X
X
X
1
5
1
√
(a)
(b)
(c)
sin
n
4 +3
n
n n2 + 1
n=1
n=1
n=1
7. Determine the convergence or the divergence of the following series using the Ratio or the
Root Test:
n
∞
∞
∞
∞
X
X
X
X
(−1)n−1 n3
2n 3n
n+1
(−3)n
(b)
(c)
(d)
(a)
n
n
n!
n!
n2
n=1
n=1
n=1
n=1
8. Find the first four non-zero terms of both the Maclaurin series and Taylor series (at x = c)
generated by the following functions:
(a) f(x) = sec x, c = π
x
, c=2
x−1
√
(c) h(x) = 1 + x, c = 3
(b) g(x) =
9. Find the radius and interval of convergence of the following series:
∞
X
(x − 1)2n−2
∞
X
(x + 4)n
(a)
n=1
n(3n )
(b)
n=1
(2n − 1)!
∞
X
(−1)n−1 (3x − 1)n
(c)
n2
n=1
∞
X
xn
(d)
n=1
nn
10. Find the Maclaurin series for the following functions:
2x
1 + x2
√ 5x
(b) g(x) = cos
(a) f(x) =
(c) h(x) = ln[(1 − x)(1 − 2x)]
(d) m(x) = sinh x
(e) n(x) = ln(1 + x 2 )
(f) p(x) = sec x
11. Use the Maclaurin series for ln(1 − x) to approximate ln
is less that 10−4 .
3
2
so that the absolute value error
12. Given that
arctan x = x −
x3 x3 x5
x 2n−1
+
−
+ · · · + (−1)n−1
+ ...,
3
3
5
2n − 1
evaluate the following integral:
Z 1
(a)
0
arctan(x 2 )
dx
x
Z 641
(b)
0
arctan x
√
dx
x
giving your answer such that the absolute error does not exceed 0.01.
|x| < 1,