IT training intelligent routines solving mathematical analysis with MATLAB, mathcad, mathematica and maple anastassiou iatan 2012 07 28

Intelligent Systems Reference Library

39

Editors-in-Chief
Prof. Janusz Kacprzyk
Systems Research Institute
Polish Academy of Sciences
ul. Newelska 6
01-447 Warsaw
Poland
E-mail:

For further volumes:
/>
Prof. Lakhmi C. Jain
School of Electrical and Information
Engineering
University of South Australia
Adelaide
South Australia SA 5095
Australia
E-mail:


George A. Anastassiou and Iuliana F. Iatan

Intelligent Routines
Solving Mathematical Analysis with Matlab,
Mathcad, Mathematica and Maple

123


Authors
George A. Anastassiou
Department of Mathematical Sciences
University of Memphis
Memphis
USA

Iuliana F. Iatan
Department of Mathematics and
Computer Science
Technical University of Civil Engineering
Bucharest
Romania

ISSN 1868-4394
e-ISSN 1868-4408
ISBN 978-3-642-28474-8
e-ISBN 978-3-642-28475-5
DOI 10.1007/978-3-642-28475-5
Springer Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012932490
c Springer-Verlag Berlin Heidelberg 2013
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the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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to the material contained herein.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)


Dedicated to our families.


VII

”Homines dum docent discunt.”
Seneca, Epistole 7

”Nihil est in intellectu, quod non prius fuerit in sensu.”
John Locke

”Les beaux (grands) esprits se rencontrent.”
Voltaire

”Men should be what they seem,
Or those that be not, would they might seem none.”
Shakespeare, Othello III. 3

”Science needs a man’s whole life. And even if you had two lives, they
would not be enough. It is great passion and strong effort that science
demands to men ...”
I. P. Pavlov


VIII

”Speech is external thought, and thought internal speech.”
A. Rivarol

”Nemo dat quod non habet.”
Latin expression

”Scientia nihil aliud est quam veritatis imago.”
Bacon, Novum Organon


Preface

Real Analysis is a discipline of intensive study in many institutions of higher
education, because it contains useful concepts and fundamental results in
the study of mathematics and physics, of the technical disciplines and
geometry.
This book is the first one of the kind that solves mathematical analysis
problems with all four related main software Matlab, Mathcad, Mathematica and Maple.

Besides the fundamental theoretical notions, the book contains many
exercises, solved both mathematically and by computer, using: Matlab 7.9,
Mathcad 14, Mathematica 8 or Maple 15 programming languages.
Due to the diversity of the concepts that the book contains, it is addressed
not only to the students of the Engineering or Mathematics faculties but
also to the students at the master’s and PhD levels, which study Real
Analysis, Differential Equations and Computer Science.
The book is divided into nine chapters, which illustrate the application of
the mathematical concepts using the computer. The introductory section of
each chapter presents concisely, the fundamental concepts and the elements
required to solve the problems contained in that chapter. Each chapter
finishes with some problems left to be solved by the readers of the book
and can verified for the correctness of their calculations using a specific
software such as Matlab, Mathcad, Mathematica or Maple.
The first chapter presents some basic concepts about the theory of sequences and series of numbers.
The second chapter is dedicated to the power series, which are particular cases of series of functions and that have an important role for some
practical applications; for example, using the power series we can find the
approximate values of some functions so we can appreciate the precision of
a computing method.


X

Preface

In the third chapter are treated some elements of the differentiation
theory of functions.
The fourth chapter presents some elements of Vector Analysis with applications to physics and differential geometry.
The fifth chapter presents some notions of implicit functions and extremes of functions of one or more variables.
Chapter six is dedicated to integral calculus, which is useful to solve various geometric problems and to mathematical formulation of some concepts

from physics.
Seventh chapter deals with the study of the differential equations and
systems of differential equations that model the physical processes.
The chapter eight deals with the line and double integrals. The line integral is a generalization of the simple integral and allows the understanding
of some concepts from physics and engineering; the double integral has
a meaning analogous to that of the simple integral: like the simple definite integral is the area bordered by a curve, the double integral can be
interpreted as the volume bounded by a surface.
The last chapter is dedicated to the triple and surface integral calculus.
Although it is not possible a geometric interpretation of the triple integral,
mechanically speaking, this integral can be interpreted as a mass, being
considered as the distribution of the density in the respective space. The
surface integral is a generalization of the double integral in some plane
domains, as the line integral generalizes the simple definite integral.
This work was supported by the strategic grant POSDRU/89/1.5/S/
58852, Project “Postdoctoral programme for training scientific researchers”
cofinanced by the European Social Fund within the Sectorial Operational
Program Human Resources Development 2007-2013.
The authors would like to thank Professor Razvan Mezei of Lander University, South Carolina, USA for checking the final manuscript of our book.
January 10, 2012
George Anastassiou,
Memphis
USA
Iuliana Iatan
Bucharest
Romania


Contents

1

Sequences and Series of Numbers . . . . . . . . . . . . . . . . . . . . . . .
1.1 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Convergent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1.1 Cauchy’s Test . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Divergent Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Operations on Convergent Series . . . . . . . . . . . . . . . . .
1.3 Tests for Convergence of Alternating Series . . . . . . . . . . . . . .
1.4 Tests of Convergence and Divergence of Positive Series . . . .
1.4.1 The Comparison Test I . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.2 The Root Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.3 The Ratio Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.4 The Raabe’s and Duhamel’s Test . . . . . . . . . . . . . . . . .
1.4.5 The Comparison Test II . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.6 The Comparison Test III . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Absolutely Convergent and Semi-convergent Series . . . . . . . .
1.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
3
5
7
11
14
16
16
19

22
26
29
30
31
34

2

Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Region of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Taylor and Mac Laurin Series . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Expanding a Function in a Power Series . . . . . . . . . . .
2.3 Sum of a Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41
41
49
49
60
65


XII

3

Contents

Differentiation Theory of the Functions . . . . . . . . . . . . . . . . .
3.1 Partial Derivatives and Differentiable Functions
of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 The Total Differential of a Function . . . . . . . . . . . . . .
3.1.3 Applying the Total Differential of a Function
to Approximate Calculations . . . . . . . . . . . . . . . . . . . . .
3.1.4 The Functional Determinant . . . . . . . . . . . . . . . . . . . . .
3.1.5 Homogeneous Functions . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Derivation and Differentiation of Composite Functions
of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Change of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Taylor’s Formula for Functions of Two Variables . . . . . . . . .
3.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102
119
126
143

4

Fundamentals of Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Derivative in a Given Direction of a Function . . . . . . . . . . . .
4.2 Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157
157
162

179

5

Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Derivative of Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Differentiation of Implicit Functions . . . . . . . . . . . . . . . . . . . .
5.3 Systems of Implicit Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Functional Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Extreme Value of a Function of Several Variables
Conditional Extremum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187
187
193
203
209

Terminology about Integral Calculus . . . . . . . . . . . . . . . . . . .
6.1 Indefinite Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1 Integrals of Rational Functions . . . . . . . . . . . . . . . . . . .
6.1.2 Reducible Integrals to Integrals of Rational
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2.1 Integrating Trigonometric Functions . . . . . . .
6.1.2.2 Integrating Certain Irrational Functions . . .
6.2 Some Applications of the Definite Integrals in Geometry
and Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 The Area under a Curve . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 The Area between by Two Curves . . . . . . . . . . . . . . . .

6.2.3 Arc Length of a Curve . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.4 Area of a Surface of Revolution . . . . . . . . . . . . . . . . . .
6.2.5 Volumes of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.6 Centre of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245
245
245

6

71
71
71
83
90
93
99

214
229

251
251
252
260
260
265
269
274

276
277


Contents

6.3 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Integrals of Unbounded Functions . . . . . . . . . . . . . . . .
6.3.2 Integrals with Infinite Limits . . . . . . . . . . . . . . . . . . . . .
6.3.3 The Comparison Criterion for the Integrals . . . . . . . .
6.4 Parameter Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7

8

Equations and Systems of Linear Ordinary
Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Successive Approximation Method . . . . . . . . . . . . . . . . . . . . . .
7.2 First Order Differential Equations Solvable
by Quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 First Order Differential Equations with Separable
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 First Order Homogeneous Differential Equations . . .
7.2.3 Equations with Reduce to Homogeneous
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.4 First Order Linear Differential Equations . . . . . . . . . .
7.2.5 Exact Differential Equations . . . . . . . . . . . . . . . . . . . . .
7.2.6 Bernoulli’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.7 Riccati’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2.8 Lagrange’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.9 Clairaut’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Higher Order Differential Equations . . . . . . . . . . . . . . . . . . . . .
7.3.1 Homogeneous Linear Differential Equations
with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . .
7.3.2 Non-homogeneous Linear Differential Equations
with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . .
7.3.2.1 The Method of Variation of Constants . . . . .
7.3.2.2 The Method of the Undetermined
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.3 Euler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3.4 Homogeneous Systems of Differential Equations
with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . .
7.3.5 Method of Characteristic Equation . . . . . . . . . . . . . . .
7.3.6 Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Non-homogeneous Systems of Differential Equations
with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Line and Double Integral Calculus . . . . . . . . . . . . . . . . . . . . . .
8.1 Line Integrals of the First Type . . . . . . . . . . . . . . . . . . . . . . . .
8.1.1 Applications of Line Integral of the First Type . . . . .
8.2 Line Integrals of the Second Type . . . . . . . . . . . . . . . . . . . . . .
8.3 Calculus Way of the Double Integrals . . . . . . . . . . . . . . . . . . .

XIII

280
280
288
293

296
302

317
317
320
322
324
326
330
332
339
340
342
346
349
349
357
357
360
367
369
371
373
377
380
395
395
396
406

421


XIV

9

Contents

8.4 Applications of the Double Integral . . . . . . . . . . . . . . . . . . . . .
8.4.1 Computing Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.2 Mass of a Plane Plate . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.3 Coordinates the Centre of Gravity of a Plane
Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4 Moments of Inertia of a Plane Plate . . . . . . . . . . . . . .
8.4.5 Computing Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5 Change of Variables in Double Integrals . . . . . . . . . . . . . . . . .
8.5.1 Change of Variables in Polar Coordinates . . . . . . . . . .
8.5.2 Change of Variables in Generalized Polar
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 Riemann-Green Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

430
430
431
433
436
438
441

441

Triple and Surface Integral Calculus . . . . . . . . . . . . . . . . . . . .
9.1 Calculus Way of the Triple Integrals . . . . . . . . . . . . . . . . . . . .
9.2 Change of Variables in Triple Integrals . . . . . . . . . . . . . . . . . .
9.2.1 Change of Variables in Spherical Coordinates . . . . . .
9.2.2 Change of Variables in Cylindrical Coordinates . . . . .
9.3 Applications of the Triple Integrals . . . . . . . . . . . . . . . . . . . . .
9.3.1 Mass of a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Volume of a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 Centre of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.4 Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Surface Integral of the First Type . . . . . . . . . . . . . . . . . . . . . .
9.5 Surface Integral of the Second Type . . . . . . . . . . . . . . . . . . . .
9.5.1 Flux of a Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5.2 Gauss- Ostrogradski Formula . . . . . . . . . . . . . . . . . . . .
9.5.3 Stokes Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

475
475
477
477
480
485
485
493
499
509
512

520
524
529
540
551

445
447
454

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579


1

Sequences and Series of Numbers

1.1

Cauchy Sequences

Definition 1.1 (see [41], p.13). A sequence (xn )n∈N is convergent if (∃)
a ∈ R such that (∀) ε > 0, (∃) nε ∈ N such that |xn − a| < ε, (∀) n nε .
Definition 1.2 (see [41], p.23). A sequence (xn )n∈N is called a Cauchy
sequence if the terms of the sequence, eventually all become arbitrarily
close to one another, i.e. if for (∀) ε > 0, (∃) nε ∈ N such that (∀) n nε
one obtains |xn+p − xn | < ε, (∀) p ∈ N.
Example 1.3. Prove that the following sequence is a Cauchy sequence :

cos n!
cos 1! cos 2!
+
+ ··· +
, (∀) n ∈ N∗
1·2
2·3
n (n + 1)
cos α cos 2α
cos nα
b) xn =
+
+ ···+
, (∀) n ∈ N∗ , (∀) α ∈ R.
12
22
n2
a) xn =

Solutions.
a) We shall have
G.A. Anastassiou and I.F. Iatan: Intelligent Routines, ISRL 39, pp. 1–40.
c Springer-Verlag Berlin Heidelberg 2013
springerlink.com


2

1 Sequences and Series of Numbers
n+p

|xn+p − xn | =
k=n+1
n+p

=
k=n+1

n+p

cos k!
k (k + 1)

k=n+1

1
1
−
k k+1

1
k (k + 1)

1
1
−
n+1 n+2

=

1
1
1
1
−
+ ··· +
−
n+2 n+3
n+p n+p+1
1
1
1
−
<
−→ 0.
=
n+1 n+p+1
n + 1 n−→∞
+

1
,
We shall use the definition of convergence of the sequence yn = n+1
∗
n ∈ N in 0 to determine the rank nε ∈ N such that (∀) n nε we have
|xn+p − xn | < ε, (∀) p ∈ N.
1
Hence, using the Definition 1.1, (∀) ε > 0, (∃) nε ∈ N such that n+1
< ε,
(∀) n nε .

It results that n > 1ε − 1; we choose

nε =

1
− 1 + 1;
ε

we achieve
n>

1
1
− 1 + 1 > − 1.
ε
ε

As (∀) ε > 0, (∃) nε = 1ε − 1 + 1 ∈ N, such that (∀) n nε we have
|xn+p − xn | < ε; according to the Definition 1.2 it results that
n

xn =
k=1

cos k!
, n ∈ N∗
k (k + 1)

is a Cauchy sequence.
b) We can write

n+p

|xn+p − xn | =
k=n+1
n+p

=
k=n+1

cos kα
k2

n+p

k=n+1

1
1
−
k−1 k

=

1
≤
k2

n+p

k=n+1

1
k (k − 1)

1
1
−
n n+1

1
1
1
1
−
···+
−
n+1 n+2
n+p−1 n+p
1
1
1
<
−→ 0.
= −
n n+p
n n−→∞
+


1.2 Fundamental Concepts

3

We shall use the definition of convergence of the sequence yn = n1 , n ∈
N in 0 to determine the rank nε ∈ N such that (∀) n
nε we have
|xn+p − xn | < ε, (∀) p ∈ N.
Hence, using the Definition 1.1, (∀) ε > 0, (∃) nε ∈ N such that n1 < ε,
(∀) n nε . It results that n > 1ε ; we choose
∗

1
+ 1;
ε

nε =
we achieve
n>

1
1
+1> .
ε
ε

As (∀) ε > 0, (∃) nε = 1ε + 1 ∈ N, such that (∀) n
nε we have
|xn+p − xn | < ε; according to the Definition 1.2 it results that
n

xn =
k=1

cos kα
, n ∈ N∗
k2

is a Cauchy sequence.

1.2

Fundamental Concepts

1.2.1

Convergent Series

Definition 1.4 (see [41], p. 29). A number series
∞

an = a1 + a2 + · · · + an + · · · ,

an =
n=1

(1.1)

n≥1

is called convergent if the sequence of its partial sums

n

Sn =

ai

(1.2)

i=1

has the finite limit

S = lim Sn .
n→∞

(1.3)

Definition 1.5 (see [41], p. 30). The quantity from (1.3) is called the sum
of the series n≥1 an .
Example 1.6. Prove that the following series are convergent:


4

1 Sequences and Series of Numbers

1
1
1
+

+ ···+
+ ···
2
5 45
16n − 8n − 3
1
√
√
b)
.
n
+
2
n
+ 2+1
n≥1
a)

Solutions.
a) We can notice that
16n2 − 8n − 3 = 16n2 + 4n − 12n − 3 = (4n − 3) (4n + 1) ;
therefore
n

Sn =

n

ak =
k=1

1
=
4

k=1

1
1
=
(4k − 3) (4k + 1)
4

1
1−
4n + 1

n

k=1

1
1
−
4k − 3 4k + 1

1
,
→
n→∞ 4

i.e.
n≥1

16n2

1
− 8n − 3

is convergent.
We can also notice that using the following Matlab 7.9 sequence:
>>syms n
>> symsum(1/(16*nˆ2-8*n-3),n,1,inf )
ans =
1/4
or in Mathcad 14:
∞
n=1

16n2

1
1
→
− 8n − 3
4

or using Mathematica 8:
ln[1]:=Sum[1/(16*nˆ2 - 8*n - 3), {n, 1, Infinity}]
Out[1]= 41

or with Maple 15:


1.2 Fundamental Concepts

5

b) We have
n

Sn =

n

ak =
k=1
n

=
k=1

=

k=1

√
k+ 2

1
√

k+ 2+1

1
1
√ −
√
k+ 2 k+ 2+1

1
1
1
√ −
√ →
√ ;
n→∞
1+ 2 n+1+ 2
1+ 2

hence
n≥1

n+

√
2

1
√
n+ 2+1

is convergent.
In Matlab 7.9 we shall have:
>>syms n
>> symsum(1/((n+sqrt(2))*(n+sqrt(2)+1)),n,1,inf )
ans =
1/(1+2ˆ(1/2))
or in Mathcad 14:
∞
n=1

√
n+ 2

√
1
√
simplify → 2 − 1
n+ 2+1

or with Mathematica 8:
ln[2]:=Sum[1/((n
√ + Sqrt[2])*(n + Sqrt[2] + 1)), {n, 1, Infinity}]
Out[2]=-1 + 2
or with Maple 15:

1.2.1.1

Cauchy’s Test

Proposition 1.7 (see [41], p. 36). The necessary and sufficient condition

for the convergence of the series n≥1 an is: (∀) ε > 0, (∃) nε ∈ N such
that:
|an+1 + · · · + an+p | < ε, (∀) n nε , (∀) p ∈ N.


6

1 Sequences and Series of Numbers

Example 1.8. Use the Cauchy’s test for testing the convergence of the
series
a)
n≥1

cos nx
, (∀) x ∈ R
5n

sin n2 x
, (∀) x ∈ R.
n3

b)
n≥1

Solutions.
a) We have
n+p

|an+1 + · · · + an+p | =

k=n+1

=
=

1
5n+1

cos kx
≤
5k
1+

1

·

5n+1

n+p

k=n+1

1
1
+ ···+ p
5
5

1

5p − 1
1
5 −1

=

1
1
· 1− p
4 · 5n
5

Therefore, (∀) ε > 0, (∃) nε ∈ N such that
5n >

1
5k

1
5n

<

< ε, (∀) n

1
→ 0.
5n n→∞
nε , i.e.

ln ε
1
=⇒ n > −
;
ε
ln 5

ln ε
it results that (∀) ε > 0, (∃) nε = − ln
5 + 1 ∈ N, such that (∀) n
we have |an+1 + · · · + an+p | < ε, i.e.

n≥1

cos nx
, (∀) x ∈ R
5n

is convergent.
b) We achieve:
n+p

|an+1 + · · · + an+p | =
k=n+1
n+p

≤
k=n+1
n+p

=
k=n+1

sin k 2 x
≤
k3

1
≤
k2

n+p

k=n+1

1
1
−
k−1 k

n+p

k=n+1

1
k3

1
k (k − 1)
=

1
1
1
−
<
→ 0.
n n+p
n n→∞

nε


1.2 Fundamental Concepts

Hence, (∀) ε > 0, (∃) nε ∈ N such that
n>
it results that (∀) ε > 0, (∃) nε =
have |an+1 + · · · + an+p | < ε, i.e.

n≥1

1
ε

1
n

< ε, (∀) n

7

nε , i.e.

1
;
ε
+ 1 ∈ N, such that (∀) n

nε we

sin n2 x
, (∀) x ∈ R
n3

is convergent.

1.2.2

Divergent Series

Definition 1.9 (see [41], p. 30). If the limit limn→∞ Sn does not exist (or
it is infinite), the series is then called divergent.
Proposition 1.10 (see [15], p. 93). If the terms of a series sequence is not
convergent to 0, the series is divergent, namely if for the series n≥1 an
we have
lim an = a = 0, ‘

n→∞

then

n≥1

an diverges.

Example 1.11. Prove that the following series are divergent:
0, 07 + 3 0, 07 + · · · +
1
√
√
b)
2n + 1 − 2n − 1
n≥1

a) 0, 07 +

c)
n≥1

d)

2n + 3n
2n+1 + 3n+1
ln

n≥1

e)
n≥1

n+1
n

a
n sin , a = 0.
n

Solutions.
a) As
lim

n

n→∞

we deduce that

n≥1

1

0, 07 = lim (0, 07) n = 1 = 0
n→∞
√
n
0, 07 diverges.

n

0, 07 + · · ·


8

1 Sequences and Series of Numbers

We can prove that using Matlab 7.9:
>> syms n
>> limit(0.07ˆ(1/n),inf )
ans =
1
or in Mathcad 14:

or with Mathematica 8:

or in Maple 15:

b) Since
1
√
lim √
= lim
n→∞
2n + 1 − 2n − 1 n→∞

√
√
2n + 1 + 2n − 1
= ∞,

2n + 1 − 2n + 1

using the Definition 1.9 it results that the series diverges.
We can also deduce that using Matlab 7.9:
>>syms n
>> limit(1/(sqrt(2*n+1)-sqrt(2*n-1)),inf )
ans =
Inf
or in Mathcad 14:

or with Mathematica 8:


1.2 Fundamental Concepts

9

or in Maple 15:

c) One obtains
3n
2n + 3n
lim n+1
=
lim
n→∞ 2
+ 3n+1 n→∞ 3n+1

2n
3n + 1

2n+1
3n+1 +

=
1

1
= 0;
3

we have used that
⎧
⎪
⎪
⎨

0,
if q ∈ (−1, 1)
1,
if q = 1
lim q =
∞,
if
q>1
n→∞
⎪
⎪
⎩
it doesn’t exists, if q ∈ (−∞, −1] .
n

Hence, based on the Proposition 1.10 it follows that the series diverges.
It will also result that in Matlab 7.9:
>> syms n
>> limit((2ˆn+3ˆn)/(2ˆ(n+1)+3ˆ(n+1)),inf )
ans =
1/3
or in Mathcad 14:

or with Mathematica 8:


10

1 Sequences and Series of Numbers

or in Maple 15:

d) We have
n+1
= 0,
n
namely we can say nothing about the series nature.
As
lim ln

n→∞

n

Sn =

ln
k=1

= ln

2
3
n+1
k+1
= ln + ln + · · · + ln
k
1
2
n

2 3
n+1
· · ····
1 2
n

= ln (n + 1)

and
lim Sn = ∞

n→∞

one deduces that the series is divergent.
The same result can be achieved using Matlab 7.9:
>> syms n k
>> limit(symsum(log((k+1)/k),k,1,n),n,inf )
ans =
Inf
or Mathcad 14:

or with Mathematica 8:


1.2 Fundamental Concepts

11

or in Maple 15:

e) It will result:
lim n · sin

n→∞

sin a
a
= lim a a n = a = 0,
n n→∞
n

therefore the series is divergent.
We can also obtain that using Matlab 7.9:

>> syms n a
>> limit(n*sin(a/n),inf )
ans =
a
or Mathcad 14:

or with Mathematica 8:

or in Maple 15:

1.2.3

Operations on Convergent Series

Proposition 1.12 (see [15], p. 95). Let
n≥1 an and
n≥1 bn be two
convergent series, which have the sums A and B respectively and let be
α ∈ R. Then:


12

1 Sequences and Series of Numbers

a) the series n≥1 αan converges and it has the sum αA, namely a convergent series may be multiplied term by term by any number α;
b) the sum (difference) of the two convergent series is a convergent series
n≥1 (an ± bn ) and it has the sum A ± B.
Example 1.13. Use the operations on convergent series to compute
the sum of the series:

a)
n≥1

b)
n≥1

2n + 3
n (n + 1) (n + 2)
2n + 3n+1 − 6n−1
.
12n

Solutions.
a) One can notice that
3
1
1
2n + 3
=
−
−
n (n + 1) (n + 2)
2n n + 1 2 (n + 2)
2
1
1
1
+
−

−
=
2n 2n n + 1 2 (n + 2)
1
1
1
1
1 1
1
=
−
+ −
=
−
2n 2 (n + 2) n n + 1
2 n n+2

an =

+

1
1
−
n n+1

;

hence

n≥1

2n + 3
=
n (n + 1) (n + 2)

n≥1

1
2

1
1
−
n n+2

+
n≥1

1
1
−
n n+1

.

We shall denote
⎧
⎪
⎨ un = αan =

⎪
⎩

vn =

1
n

−

1
2

1
n

−

1
n+1 .

1
n+2

One can notice that
n

n

ai =

i=1

i=1

1
1
−
i
i+2

1
1
1
1
1
1
1 1 1
+ − + ···+
− +
−
+ −
3 2 5
n−2 n n−1 n+1 n n+2
1
1
1
−
= 1+ −
2 n+1 n+2

= 1−


1.2 Fundamental Concepts

13

and
lim

n→∞

1
1
1
−
−
2 n+1 n+2

1+

hence it results that the series

=

3
;
2

an converges.

n≥1

Applying the Proposition 1.12 it follows the series
converges and
n

αai =
i=1

n≥1

αan will also

3
1 3
· = .
2 2
4

Similarly,

n

n

1
1
−
i

i+1

vi =
i=1

i=1

= 1−

=1−

1
1
1 1 1
+ − + ··· + −
2 2 3
n n+1

1
n+1

and
lim

n→∞

1−

hence, it results that the series

1
n+1

n≥1 vn

= 1;

converges.

Applying the Proposition 1.12 it follows that the series
will also converge and it has the sum
U +V =

n≥1

(un + vn )

7
3
+1= .
4
4

b) We shall have

n≥1

2n + 3n+1 − 6n−1
=
12n

n≥1

1
6
an

It results that

n

+3
n≥1

1
4
bn

n

−

1
6

n≥1

1
2
cn

n

.