Lesson 1: Series and Sequences Study Notes
Official Business - MAPÚA UNIVERSITY
SENIOR HIGH SCHOOL - MATH01
General Mathematics
Course Outcome 4
Lesson 1: Series and Sequences
Source: shs.mapua.edu.ph
Lesson Objectives
At the end of this lesson, students should be able to:
Illustrate and differentiate a series from a sequence.
Use the sigma notation to represent a series.
Illustrate arithmetic sequence, geometric sequence, and harmonic sequence.
Illustrate and apply the principle of mathematical induction in proving identities.
Lesson Outline
1.1. Definition of Series and Sequences
1.2. The Sigma Notation
1.3. Arithmetic, Geometric, and Harmonic Series and Sequence
1.1. Definition of Series and Sequences
A sequence is defined as a function whose domain is the set of positive integers or the set:
ext{Domain} = ext{set of positive integers} = \{1, 2, 3, \, ext{…}\}
A sequence is a list of numbers separated by commas.
Example: The sequence a_n = ext{1, } -1, rac{1}{2}, rac{1}{3}, - rac{1}{4}
A series represents the sum of the terms of a sequence.
Thus, a series is defined as a sum of numbers separated by “+” or “-” sign.
Example: The series S = 1 - rac{1}{2} + rac{1}{3} - rac{1}{4} = rac{7}{12}
The sequence with the nth term is usually denoted by a_n, and the associated series is given by
S = 1 + 2 + 3 + …
1.1. Definition of Series and Sequences - Examples
Example 1.1.1. Determine the first five terms of each defined sequence and give their associated series.
a_n = 2 - n
b_n = 1 + 2 + 3^n
c_n = (-1)^n
d_n = n
Solution:
Sequence 1: a_n = 2 - n
First five terms:
a_1 = 2 - 1 = 1
a_2 = 2 - 2 = 0
a_3 = 2 - 3 = -1
a_4 = 2 - 4 = -2
a_5 = 2 - 5 = -3
Associated series:
S_a = 1 + 0 - 1 - 2 - 3 = -5
Sequence 2: b_n = 1 + 2 + 3^n
First five terms:
b_1 = 1 + (2 imes 1) + 3(1^2) = 6
b_2 = 1 + (2 imes 2) + 3(2^2) = 17
b_3 = 1 + (2 imes 3) + 3(3^2) = 34
b_4 = 1 + (2 imes 4) + 3(4^2) = 57
b_5 = 1 + (2 imes 5) + 3(5^2) = 86
Associated series:
S_b = 6 + 17 + 34 + 57 + 86 = 200
Sequence 3: c_n = (-1)^n
First five terms:
c_1 = (-1)^1 = -1
c_2 = (-1)^2 = 1
c_3 = (-1)^3 = -1
c_4 = (-1)^4 = 1
c_5 = (-1)^5 = -1
Associated series:
S_c = -1 + 1 - 1 + 1 - 1 = -1
Sequence 4: d_n = n
First five terms:
d_1 = 1
d_2 = 1 + 2 = 3
d_3 = 1 + 2 + 3 = 6
d_4 = 1 + 2 + 3 + 4 = 10
d_5 = 1 + 2 + 3 + 4 + 5 = 15
Associated series:
S_d = 1 + 3 + 6 + 10 + 15 = 35
1.2. The Sigma Notation
Sigma notation is utilized by mathematicians to denote a sum.
The uppercase Greek letter Σ (sigma) indicates summation.
The sigma notation consists of various components or parts such as the index of summation, the lower limit, the upper limit, and the expression to be summed.
Example 1.2.1. Expand each summation and simplify if possible.
egin{aligned}& ext{Expand: } \ & ext{(a) } \sum{k=2}^{4} (2 + 3)\ ext{(b) } \sum{n=0}^{5} (2)\ ext{(c) } \sum{i=1}^{3} i\ ext{(d) } \sum{m=1}^{6} rac{1}{
ho + 1} \ ext{(e) } … \ ext{and other linear combinations (as shown in the original examples).} \ ext{The following representations illustrate how to interpret and calculate sums in sigma notation.} ext{Resolve them appropriately by any method mentioned before.} ext{Depending on the range of terms and value of isturions used, the results will vary.”}
Example 1.2.2. Write each expression in sigma notation.
1) 1 + rac{1}{2} + rac{1}{3} + rac{1}{4} + … + rac{1}{100}
- ext{This can be represented as } \ ext{(i)}
- ext{Write: } \ ext{(a)} ext{ } \sum{i=1}^{100} rac{1}{i} 2) -1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 - 9 + … - 25 - ext{This can be represented as } \ ext{(i)}\sum{i=1}^{25} (-1)^i imes i
3) 2 + 4 + 6 + 8 + … + 20
- ext{This can be represented as } \ ext{(i)}\sum{i=1}^{10} 2i 4) 1 + rac{1}{2} + rac{1}{4} + rac{1}{8} + rac{1}{16} + rac{1}{32} + rac{1}{64} + rac{1}{128} - ext{This can be represented as } \ ext{(i)}\sum{i=0}^{7} rac{1}{2^i}
Properties of Sigma Notation
The sum of a finite arithmetic sequence can be deduced using the formula for an arithmetic series.
egin{aligned} ext{Let } \Sn = ext{the sum of the first n terms of any sequence.} \ Sn = rac{n}{2}(a + l)\ ext{where}\ a = ext{first term}\ l = ext{last term} \ ext{For the series from 1 to n, this resolves to:} \ \ S_n = rac{n(n + 1)}{2} \ ext{thus applied to certain limits.} ext{This will assist in determining expected outcomes and values in implementations.} ext{Summation of series will produce distinct results depending on individual limits applied with various functions.}
ext{If } (c)\sum{i=a}^{b} c(i) = c\sum{i=a}^{b} 1
For any real number c, \ If a term is uniformly treated and operated across the sigma, this could yield unique benefits if appropriate conversions are handled correctly.
\sum{i=a}^{b} (c + d) =\sum{i=a}^{b} c + \sum_{i=a}^{b} d\
1.3. Arithmetic, Geometric, and Harmonic Series and Sequences
Arithmetic Sequence
An arithmetic sequence is one in which each term after the first is obtained by adding a constant (called the common difference) to the preceding term.
Formula for the nth term:
an = a1 + (n-1)d
Where:
a_n is the nth term,
a_1 is the first term,
d is the common difference.
The associated arithmetic series with n terms can be expressed as:
Sn = rac{n}{2} (a1 + a_n)
or equivalently:
Sn = rac{n}{2} (2a1 + (n-1)d)
Geometric Sequence
A geometric sequence is defined such that after the first term, each term is found by multiplying the preceding term by a fixed constant (known as the common ratio).
Formula for the nth term:
gn = g1 imes r^{(n-1)}
Where:
g_n is the nth term,
g_1 is the first term,
r is the common ratio.
The associated geometric series with n terms can be expressed as:
Sn = g1 rac{1 - r^n}{1 - r} \ ext{(if r ≠ 1)}
For an infinite geometric series where |r| < 1, the sum is given by:
S = rac{g_1}{1 - r}
Harmonic Sequence
A harmonic sequence is defined such that each term is the reciprocal of an arithmetic sequence.
If the arithmetic sequence has a n-th term:
a_n, then the n-th term of the harmonic sequence is given by:
hn = rac{1}{an}
To convert it for practical solving, take the reciprocals of the arithmetic values to yield harmonic results in scenarios requiring such manipulations.
Examples on Series and Sequences
Identify the nature of given terms; indicate whether they are Series (SER) or Sequence (SEQ). (Referential structure meant for sequence identification reconnects to earlier definitions.)
a) $1, 2, 4, 8, …$ - SEQ
b) $2, 8, 10, 18, …$ - SEQ
c) $−1 + 1−1 + 1−1$ - SER
d) $1/2, 2/3, 3/4, 4/5, …$ - SEQ
e) $1 + 2 + 2^2 + 2^3 + 2^4$ - SER
f) $1 + 0.1 + 0.001 + 0.0001$ - SER
For the nature of sequences aligned by arithmetic (A), geometric (G), harmonic (H), or others (O):
a) $3, 5, 7, 9, 11, …$ - A (common difference is 2)
b) $2, 4, 9, 16, 25, …$ - O
c) $1/4, 1/16, 1/64, 1/256, …$ - G
d) $1/3, 2/9, 3/27, 4/81, …$ - O
e) $1/5, 1/9, 1/13, 1/17, 1/21, …$ - H (common difference of the reciprocals is 4)
f) $ ext{Set of square roots: } √3, √4, √5, √6, …$ - O
g) $ ext{Set of decreasing decimals: } 0.1, 0.01, 0.001, 0.0001,…$ - G (common ratio of $ rac{1}{10}$)
Series identification and determination of sum:
(a) $4 + 9 + 14 + … + 64$ - Arithmetic, $442$
(b) $81 + 27 + 9 + … + 1$ - Geometric, $ rac{9841}{81}$
(c) $−10 - 2 + 6 + … + 46$ - Arithmetic, $144$
(d) $10 + 2 + 0.4 + 0.08 + …$ - Infinite Geometric, $12.5$
(e) $1 - 0.1 + 0.01 - 0.001 + …$ - Infinite Geometric, $10 rac{11}{10}$
Conclusion
Understanding sequences and series entails grasping how they are formed and manipulated through algebraic functions and sigma notation.
This knowledge lays the groundwork for more advanced mathematical concepts and their applications in various scientific fields.
Acknowledgments
Thank you!
Source: MAPÚA UNIVERSITY