1. Look carefully at the images ( I and I ) of objects (sticks) arranged sequen-
1
2
tially.
Both are meant to represent the first few terms of an infinite sequence of numbers
African Virtual University 71
{ a . In each case the arrow points in the direction of increasing values of the
n }
index n of the elements of the sequence.
Questions
(i) Is the sequence { a represented by the Image I increasing or de-
n }
1
creasing?
(ii) Is the sequence converging or not converging (diverging)?
Pose and attempt to answer the same two questions for the sequence represented
by the image I .
2
2. Consider the sequence of numbers whose first five terms are shown here:
1
1
1
1
1, , , , , …
2
3
4
5
African Virtual University 72
Key Definitions
A clear understanding by the learner, of terms used in this learning activity, is
very essential. The learner is therefore strongly advised to learn and carefully
note the following definitions.
(i) Sequences
• A sequence { a is said to be monotone if it is increasing, decreasing, non-
n }
increasing or non-decreasing.
• A sequence { a is said to be bounded if there exist two numbers m, M
n }
such that m ≤ a
M for all n larger than some integer N .
n ≤
(ii) Series
∞
• Absolute convergence: A series ∑ a is said to converge absolutely if the
k
k =1
∞
series ∑ a converges.
k
k =1
∞
• Conditional convergence: If the series ∑ a converges but the series
k
k =1
∞
∞
∑ a diverges, then a is said to converge conditionally.
k
∑ k
k =1
k =1
∞
• Positive term series: If a
for all n , then the series
a is called
n > 0
∑ k
k
a positive term series.
=1
• Alternating series: If a
for all n then the series
n > 0
∞
∑(− )k+1
1
a
a
a
a
a
k =
.....
1 −
2 +
3 −
4 +
k=1
or
∞
∑(− )
1 k a
a
a
a
a
k = −
.....
1 +
2 −
3 +
4 −
k=1
is called an alternating series.
African Virtual University 73
Mathematical Analysis
The principal reference text for the learner in the topic of Sequences and Series
is S.G. Gill. The material is contained in Chapter 8 (pp 315 – 354)
Sequences
For a sequence, the mathematical problem is
Either: to determine whether it converges or diverges,
or: to find the limit if it converges.
After giving a mathematical definition of the concept of convergence of a se-
quence, a set of standard algebraic properties of convergent sequences is given
on page 316. The set includes limits of sums, differences, products and quotients
of convergent sequences. The learner is urged to know those properties and read
carefully the proofs that immediately follow (pp 316 – 319).
Conditions for convergence of a sequence are very well spelt out in the main
reference sighted above.
Do THIS
Study carefully the set of key definitions given on page 320 of the text. The
definitions are immediately followed by a key theorem which gives criteria for
convergence. In a group work, go over the given proofs to ensure you understand
and accept the premises (conditions stated) and the conclusion of the theorem.
Error Alert: Please note an unfortunate printing error that appears on page 323
n
of the reference, where the t
n h partial sum ∑ t is referred to as an infinite
k
k =1
∞
series. This should be corrected to ∑ t .
k
k =1
Do THIS
On the basis of the definition of convergence given on page 315, the properties
of convergent sequences given on page 316 and the definitions of terms and
convergence criteria stated and proved starting from page 320, solve as many
problems as you can from Exercise 8.1 (pp 326 – 327).
African Virtual University 74
infinite series
∞
In general, infinite series are derived from sequences. An infinite series ∑ a can
k
k =1
be viewed as a sum of the terms of a sequence {a and convergence of the series
k }
is defined in terms of convergence of the associated sequence {S , S , S , S
1
2
3
4
}
,...
n
of the series, where S
a is the sum of the first n terms of the series, also
n = ∑ k
k
=1
called the n – th partial sum.
Example
⎧
1
⎫
Let ⎨
⎬ be a sequence of numbers from which we can form the se-
⎩ (
n n + )
1 ⎭
∞
1
ries ∑
. To determine whether or not this series converges, we need
k k
k
(
)
1
1
+
=
n
1
1
1
1
to form its n – th partial sum S
. Since
=
−
n = ∑ (
)
1
k(k + )
1
k k +1
1 k k
k
+
=
n
1
⎛ 1
1 ⎞
1
o n e f i n d s S
, a n d s i n c e
n =
=
∑
∑⎜ −
⎟ = 1−
k 1 k(k
k k
n
=
+ )
1
⎝
+1⎠
+1
⎛
1
now lim S
, we conclude that the infinite series
n = lim 1
⎞
⎜ −
⎟ = 1
→
n ∞
→
n ∞⎝
n +1⎠
∞
1
∑
converges, and that its sum is unity.
k k
k
(
)
1
1
+
=
Presentation of the material on infinite series is done in four major stages.
• Section 8.4 (pp 329 – 334) is devoted to positive term series
Under positive term series some very important theorems are stated, including:
(i) The comparison test (page 339)
(ii) The ratio test (page 331) and
(iii) The n – th root test.
These are stated and quite elegantly proved.
African Virtual University 75
Do THIS
Study carefully the assumptions made for each of the tests, read carefully the
proofs given and then check the level of your understanding by solving as pro-
blems as you can in Exercise 8.2 (pp 335 – 341).
• Section 8.5 (pp 341 - 346) presents alternating series
Alternating series are quite common when evaluating some power series at
a fixed point. Important results related to alternating series include the twin
concepts of absolute and conditional convergence associated with such series.
A classical example of an alternating series which is conditionally convergent
k +1
∞
1
is the series ∑ (− )
1
which can be shown to converge. However, the series
k
=1
k
obtained by taking absolute values of the terms is the harmonic series which we
know diverges. Its being divergent can easily be proved using the integral test.
• Section 8.6 (pp 347 – 354) deals with power series
A power series is a series of the form
∞
∞
k
a x
∑
or
k
a (x − c) , where c is a constant.
k
∑ k
k=1
k =1
The convergence of a power series can be established by applying the ratio
test. Here one considers
n
a x +1
a
a
n+1
lim
=
n
x
1
lim
+
or
n
x − c
1
lim
+
.
n
n→ ∞
a x
n→ ∞ a
n→∞ a
n
n
n
a
If
n+1
lim
= L then for convergence one demands that x L < 1 or
n→ ∞ a
n
x − c L < 1. In turn, these conditions lead to the following restrictions on the
possible values of x for which the power series will converge:
African Virtual University 76
1
1
x < or x − c <
.
L
L
Each of these inequalities defines an interval within which x must lie for the
power series to converge. The intervals are:
1
1
1
1
−
< x <
or c −
< x < c +
L
L
L
L
These are known as intervals of convergence of the respective power series and
1
1
the quantities R =
or R = c +
( L ≠ 0 ) are called radius of conver-
L
L
gence.
Question: If L = 0 , for what values of x do the series converge? And what
happens if L = ∞ ?
Section 8.7 (pp 354 – 360) presents Taylor series expansion
These are presented as special cases of power series, and all the results established
for general power series apply.
Software Activity
Check your Taylor series with mxMaxima.
For example:
Type taylor(x^3,x,1,4) and press RETURN
This will show the first 4 terms of a Taylor expansion of (x – 1)³
It will give:
1+3*(x-1)+3*(x-1)^2+(x-1)^3+...
Exercises (From Gill: The Calculus Bible)
Exercise 8.1
page 326
African Virtual University 77
XV.
Synthesis of the Module
Completion of Unit Four marks the end of this module. At this juncture, and before
presenting oneself for the summative evaluation, it is desirable and appropriate
that the learner reflects back on the mission, objectives, activities and attempts
to form a global picture of what he/she is expected to have achieved as a result
of the collective time and efforts invested in the learning process.
As laid out in the module overview in Section 6, the learner is expected to have
acquired knowledge of the basic concepts of differential and integral calculus of
functions of one and several independent variables. Inherently imbedded in the
key word “calculus” is the underlying concept of “limits” which underpins the
entire spectrum of coverage of the module.
The learner is now expected to be able to comfortably define the limit concept,
evaluate limits of functions, sequences and infinite series. One is also expected
to appreciate the use of the limit concept in defining the concepts of continuity,
differentiation (ordinary and partial) and integration (Riemann integral). Thus, the
signposts (salient features) on the roadmap for this module are limits, continuity,
differentiation and integration as applied to functions of both single and several
(specifically two) independent variables.
The key areas of application of the knowledge acquired are in determining local
maxima and minima, areas, volumes, lengths of curves, rates of change, moments
of inertia and centers of mass.
The module has been structured with a view to escorting and guiding the learner
through the material with carefully selected examples and references to the core
references. The degree of mastery of the module contents will largely depend on
the learner’s deliberate and planned efforts to monitor his/her progress by solving
the numerous self-exercise problems that are pointed out or given throughout
the module.
African Virtual University 78
Mathematics Module 3: Calculus
Unit 4: Calculus of Functions of Several Variables (30 Hrs)
Activity 1: Limits, Continuity and Partial Derivatives (11 Hrs)
Specific Learning Objectives
At the end of this activity the learner should be able to:
• Find out whether or not a limit exists;
• Evaluate limits of functions of several variables;
• Determine continuity of a function at a point and over an interval.
• Find partial derivatives of any order.
• Expand a function in a Taylor series about a given point
Summary
The first three units of this module dealt exclusively with the elementary dif-
ferential and integral calculus of functions of a single independent variable,
symbolically expressed by the equation y = f ( x) . However, only a very small
minority of real life problems may be adequately modeled mathematically using
such functions. Mathematical formulations of many physical problems involve
two, three or even more variables.
In this unit, and without loss of generality, we shall present the elementary dif-
ferential and integral calculus of functions of two independent variables. Speci-
fically, in this learning activity, we shall discuss the concepts of limit, continuity
and partial differentiation of functions of two independent variables.
Compulsory Reading
All of the readings for the module come from Open Source text books. This means
that the authors have made them available for any to use them without charge.
We have provided complete copies of these texts on the CD accompanying this
course.
The Calculus Bible, Prof. G.S. Gill: Brigham Young University, Maths Depart-
ment, Brigham Young University – USA.
African Virtual University 79
Internet and Software Resources
Software
Extend your expertise with mxMaxima. Investigate any functions that you find.
Check in the wxmaxima manual to find if wxMaxima can do them. If it can
practice and explore.
Wolfram Mathworld (Visited 07.11.06)
http://mathworld.wolfram.com/PartialDerivative.html
Read this entry for Partial Derivatives.
Follow links to explain specific concepts as you need to.
Wikipedia (visited 07.11.06)
http://en.wikipedia.org/wiki/Partial_derivative
Read this entry for Partial Derivatives.
Follow links to explain specific concepts as you need to.
Use Wikipedia and MathWorld to look up any key technical terms that you come
across in this unit.
Key Concepts
Domain and Range
If z = f ( x, y) is a function of two independent variables then the set D = {
2
( ,
x y) ∈ ℜ } of points for which the function z = f ( x, y) is defined is
called the Domain of z, and the set R = {z = f (x, y) : (x, y) ∈ D} of values
of the function at points in the domain is called the Range of z.
Graph of a function
The graph of a function z = f ( x, y) is the set G = {(x, y, z) ∈ ℜ : (x, y) ∈ D}
While the graph of a function of a single variable is a curve in the y – plane, the
graph of a function of two independent variables is usually a surface.
African Virtual University 80
Level curve
A level curve of a function of two variables z = f ( x, y) is a curve with an equation of the form f ( x, y) = C , where C is some fixed value of z .
Limit at a point
Let ( a, b) be a fixed point in the domain of the function z = f ( x, y) and L be a real number. L is said to be the limit of f ( x, y) as ( x, y) approaches ( a, b) written
lim
f (x, y) = L , if for every small number ∈> 0 one can find
( x, y)→(a,b)
a corresponding number δ = δ ( )
∈ such that f (x, y) − L <∈ whenever
0 < (x − 2
a) + (y − 2
)
b < δ
One also writes
lim f (x, y) = L or f ( x, y) approaches L as ( x, y) approaches ( a, b) .
x→a
y→b
Continuity
A function z = f ( x, y) is continuous at a point (a, b) ∈ D if three conditions are satisfied:
(i) f ( x, y) is defined at ( a, b) [the value f ( a, b) exists], (ii) f ( x, y) has a limit L as ( x, y) approaches ( a, b) , and (iii) The limit L and the value f ( a, b) of the function are equal.
This definition can be summed up by writing .
lim
= f (a, )
b
( x, y)→(a,b)
Partial derivative
A partial derivative of a function of several variables is the derivative of the
given function with respect to one of the several independent variables, treating
all the other independent variables as if they were real constants. For, example, if
f (x, y, z) = x2 y + 3xz2 − xyz is a function of the