The Catholic University of Malawi
Faculty Of Science
Department of Mathematical Science
Practise Topic: The Sequences and Series
Practice Questions
Composed by: KENNEDY KEN
Student ID: BEDMAT/NE/2023/1324
Date: 27 NOVEMBER, 2024
Exercises 11.2
1. (a) What is the difference between a sequence and a series? (b) What is a convergent
series? What is a divergent series?
2. Explain what it means to say that ∞
P
n=1 an = 5.
3. Calculate the sum of the series 4n=1 an , whose partial sums are given:
P
Sn = n2 − 3(0.8)n .
n2 −1
4. Sn = 4n2 +1
5. –8. Calculate the first eight terms of the sequence of partial sums correct to four
decimal places. Does it appear that the series is convergent or divergent?
P∞ 1
(a) n=1 n3
P∞ 1
(b) n=1 ln(n+1)
P∞ n√
(c) n=1 1+ n
P∞ (−1)n−1
(d) n=1 n!
6. –14. Find at least 10 partial sums of the series. Graph both the sequence of terms
and the sequence of partial sums on the same screen. Does it appear that the series
is convergent or divergent? If it is convergent, find the sum. If it is divergent,
explain why.
P∞ 12
(a) n=1 (−5)n
P∞
(b) n=1 cos n
P∞
(c) √ n
n=1 n2 +4
P∞
(d) √1
n=1 n+1
P∞ n
(e) n=1 n+2
2n
7. P
Let an = 3n+1 . (a) Determine whether {an } is convergent. (b) Determine whether
∞
a
n=1 n is convergent.
Pn P∞
8. (a) Explain
Pn the difference
Pn between i=1 a i and i=1 ai . (b) Explain the difference
between i=1 ai and j=1 aj .
9. –26. Determine whether the geometric series is convergent or divergent. If it is
convergent, find its sum.
P∞ n−1
(a) n=1 6(0.9)
P∞ n−1
(b) n=1 10(−9)
P∞ n−1
(c) n=1 (−3) /4n
P∞ πn
(d) n=1 3n+1
P∞ (−1)n
(e) n=1
√
2n
