Lecture 1 of 4MTH221: Introduction and Summary

University of Zululand

(University of Zululand) Lecture 1 of 4MTH221: Introduction and Summary 1/8

Introduction to Sequences

A sequence is an ordered list of numbers defined by a specific rule.

Sequences are denoted by terms like an , bn , and cn , where the subscript
indicates the position in the sequence.
Examples:
1 The sequence of natural numbers: an = n, starting from 1. This gives
1, 2, 3, 4, . . .
2 The arithmetic sequence: bn = 3 + 2(n − 1), where the difference between
each term is constant (2 in this case). This gives 3, 5, 7, 9, . . .
3 The geometric sequence: cn = 2 · 3n−1 , where each term is three times the
previous term. This gives 2, 6, 18, 54, . . .
4 The Fibonacci sequence: dn = dn−1 + dn−2 with d1 = 1 and d2 = 1. This
gives 1, 1, 2, 3, 5, 8, . . .

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 2/8

Sigma Notation
Sigma (Σ) notation is used to denote the sum of a sequence of terms.
General form: nk=m ak indicates the sum of terms ak from k = m to k = n.
P

The index k is a dummy variable and can be replaced without changing the
meaning of the sum.
Examples:
P5
1 The sum of the first five natural numbers: k=1 k = 1 + 2 + 3 + 4 + 5 = 15
2 The sum of the first four terms ofPan arithmetic sequence where each term
is 3k − 2 (with k starting at 1): 4k=1 (3k − 2) = 1 + 4 + 7 + 10 = 22
3 The sum P of the first three terms of a geometric sequence where each term is
2 · 3k−1 : 3k=1 2 · 3k−1 = 2 + 6 + 18 = 26
4 The
P5 sum2 of the squares of the first five positive integers:
2 2 2 2 2
k=1 k = 1 + 2 + 3 + 4 + 5 = 55

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 3/8

Properties of Sigma Notation

Pn
Pn Pn
1 Sum of Sums: k=0 a k + k=0 b k = k=0 (ak + bk )
Pn Pn
2 Scalar Multiplication: α k=0 ak = k=0 αak
Partial Sums: m
P Pn Pn
k=0 ak + k=m+1 ak = k=0 ak
3

Changing Indices: nk=j ak = n−j

P P
i=0 ai+j
4

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 4/8

Properties of Sigma Notation: Examples

1. Sum of Sums
3
X 3
X 3
X
k+ 2k = (k + 2k) = 18
k=1 k=1 k=1

This illustrates combining two sums into one.

2. Scalar Multiplication
3
X 3
X
2 k= 2k = 12
k=1 k=1

Demonstrates factoring a constant in or out of the sum.

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 5/8

Properties of Sigma Notation: Examples

3. Partial Sums
2
X 4
X 4
X
k+ k= k = 10
k=1 k=3 k=1

Shows splitting a sum into parts.

4. Changing Indices
3
X 2
X
2
k = (i + 1)2 = 14
k=1 i=0

Illustrates adjusting the index of summation.

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 6/8

Specific Cases

Pn
If all terms ak are equal to a constant r , then k=0 r = (n + 1)r .
P3
Example: For r = 5 and n = 3, k=0 5 = 20.
A
Psum of ones over any range is simply the count of terms in the range:
n
k=0 1 = n + 1.
P4
Example: For n = 4, k=0 1 = 5.

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 7/8

Conclusion

Mastering sigma notation and properties of sums is crucial for dealing with
series in calculus and mathematical analysis.
It facilitates the concise expression and manipulation of series, foundational
for topics such as series convergence tests, power series, and Fourier series.

(University of Zululand) Lecture 1 of 4MTH212: Introduction and Summary 8/8