DAMAGE BOOK
Pages are missing
00
166046
Higher Mathematics
for
Engineers and Physicists
BY IVAN S. SOKOLNIKOFF, Ph.D.
Professor of Mathematics, University of Wisconsin
AND ELIZABETH 8. SOKOLNIKOFF, Ph.D.
Formerly Instructor in Mathematics, University of Wisconsin
SECOND EDITION FOURTH IMPRESSION
McGRAW-HILL BOOK COMPANY, INC.
NEW YORK AND LONDON 1941
HIGHER MATHEMATICS FOB ENGINEERS AND PHYSICISTS
COPYRIGHT, 1934, 1941, BY TUB McGRAw-HiLL BOOK COMPANY, INC.
PRINTED IN THE UNITED STATES OF AMERICA
All rights reserved. This book, or
parts thereof, may not be reproduced
in any form without permission of
the publishers.
THE MAPLE PRESS COMPANY, YORK, PA
PREFACE
The favorable reception of the First Edition of this volume appears to have sustained the authors' belief in the need of a book on mathematics beyond the calculus, written from the point of view of the student of applied science. The chief purpose of the book is to help to bridge the gap which separates many engineers from mathematics by giving them a bird's-eye view of those mathematical topics which are indispensable in the study of the physical sciences.
It has been a common complaint of engineers and physicists that Ae usual courses in advanced calculus and differential equations place insufficient emphasis on the art of formulating physical problems in mathematical terms. There may also be a measure of truth in the criticism that many students with pro- nounced utilitarian leanings are obliged to depend on books that are more distinguished for rigor than for robust uses of mathematics.
This book is an outgrowth of a course of lectures offered by one of the authors to students having a working knowledge of the elementary calculus. The keynote of the course is the practical utility of mathematics, and considerable effort has been made to select those topics which arc of most frequent and immediate use in applied sciences and which can be given in a course of one hundred lectures. The illustrative material has been chosen for its value in emphasizing the underlying principles rather than for its direct application to specific problems that may confront a practicing engineer.
In preparing the revision the authors have been greatly aided by the reactions and suggestions of the users of this book in both academic and engineering circles. A considerable portion of the material contained in the First Edition has been rear- ranged and supplemented by further illustrative examples, proofs, and problems. The number of problems has been more than doubled. It was decided to omit the discussion of improper integrals and to absorb the chapter on Elliptic Integrals into
vi PREFACE
much enlarged chapters on Infinite Series and Differential Equations. A new chapter on Complex Variable incorporates some of the material that was formerly contained in the chapter on Conformal Representation. The original plan of making each chapter as nearly as possible an independent unit, in order to provide some flexibility and to enhance the availability of the book for reference purposes, has been retained.
I. S. S. E. S. S.
MADISON, WISCONSIN, September, 1941.
CONTENTS
PAGE
PREFACE v
CHAPTER I SECTION INFINITE SERIES
1. Fundamental Concepts 1
2. Series of Constants . ..... 6
3. Series of Positive Terms . 9
4. Alternating Series ' 15
5. Series of Positive and Negative Terms . . . 16
6. Algebra of Series . ... 21
7. Continuity of Functions Uniform Convergence . 23
8. Properties of Uniformly Convergent Series . . .28
9. Power Series . 30
10. Properties of Power Series 33
11. Expansion of Functions in Power Series ... 35
12. Application of Taylor's Formula . ... 41 J3. Evaluation of Definite Integrals by Means of Power Series ... 43
14. Rectification of Ellipse. Elliptic Integrals .47
15. Discussion of Elliptic Integrals . . . 48
16. Approximate Formulas in Applied Mathematics . . 55
CHAPTER II FOURIER SERIES
17. Preliminary Remarks . . . . 63
18. Dinchlet Conditions. Derivation of Fourier Coefficients .... 65
19 Expansion of Functions in Fourier Series . 67
20 Sine and Cosine Series . .... 73
21. Extension of Interval of Expansion 76
22. Complex Form of Fourier Series . 78
23. Differentiation and Integration of Fourier Series 80
24. Orthogonal Functions 81
CHAPTER III SOLUTION OF EQUATIONS
25. Graphical Solutions 83
26. Algebraic Solution of Cubic 86
27. Some Algebraic Theorems 92
28. Homer's Method 95
riii CONTENTS
JBCTION PAGE
29. Newton's Method . . . 97
30. Determinants of the Second and Third Order 102
31. Determinants of the nth Order. . ... . 106
32. Properties of Determinants . . . 107
33. Minors . . . . . 110
34. Matrices and Linear Dependence 114
35. Consistent and Inconsistent Systems of Equations 117
CHAPTER IV PARTIAL DIFFERENTIATION
36. Functions of Several Variables . 123
37. Partial Derivatives . 125
38. Total Differential i27
39. Total Derivatives . 130
40. Euler's Formula 136
41. Differentiation of Implicit Functions . 137
42. Directional Derivatives 143
43. Tangent Plane and Normal Line to a Surface 146
44. Space Curves 149
45. Directional Derivatives in Space 151
46. Higher Partial Derivatives 153
47. Taylor's Series for Functions of Two Variables 155
48. Maxima and Minima of Functions of One Variable 158
49. Maxima and Minima of Functions of Several Variables . 160
50. Constrained Maxima and Minima . .163
51. Differentiation under the Integral Sign 167
CHAPTER V
MULTIPLE INTEGRALS
'52. Definition and Evaluation of the Double Integral 173
53. Geometric Interpretation of the Double Integral 177
54. Triple Integrals ... 179
55. Jacobians. Change of Variable 183
56. Spherical and Cylindrical Coordinates 185
57. Surface Integrals . . 188
58. Green's Theorem in Space . . . 191
59. Symmetrical Form of Green's Theorem . . . 194
CHAPTER VI LINE INTEGRAL
60. Definition of Line Integral . . 197
61. Area of a Closed Curve 199
62. Green's Theorem for the Plane . . . 202
63. Properties of Line Integrals 206
64. Multiply Connected Regions . . . . . 212
65. Line Integrals in Space . 215
66. Illustrations of the Application of the Line Integrals . .217
CONTENTS ix
SECTION PAGE
CHAPTER VII
ORDINARY DIFFERENTIAL EQUATIONS
67. Preliminary Remarks . ... 225
68. Remarks on Solutions . ... 227
69. Newtonian Laws . . 231
70. Simple Harmonic Motion . 233
71. Simple Pendulum . . 234
72. Further Examples of Derivation of Differential Equations 239
73. Hyperbolic Functions 247 ^4. First-order Differential Equations 256 75. Equations with Separable Variables . 257 *f6. Homogeneous Differential Equations . . 259
77. Exact Differential Equations 262
78. Integrating Factors . 265
79. Equations of the First Order in Which One of the Variables Does Not Occur Explicitly . 267
80. Differential Equations of the Second Order 269
81. Gamma Functions . 272
82. Orthogonal Trajectories 277 '83. Singular Solutions . ... 279
84. Linear Differential Equations . 283
85. Linear Equations of the First Order . . 284
86. A Non-linear Equation Reducible to Linear Form (Bernoulli's Equation) . . 286
87 Linear Differential Equations of the nth Order . 287
88 Some General Theorems . . ... 291
89. The Meaning of the Operator
Z>» + oiJ)»-i + : + an-iD + anf(x) ' . . 295
90. Oscillation of a Spring and Discharge of a Condenser 299
91. Viscous Damping 302
92. Forced Vibrations . . ... 308
93. Resonance . 310
94. Simultaneous Differential Equations . . 312
95. Linear Equations with Variable Coefficients . .... 315
96. Variation of Parameters . . 318
97. The Euler Equation . . 322
98. Solution in Series . ... 325
99. Existence of Power Series Solutions . 329
100. BesseTs Equation 332
101. Expansion in Series of Bessel Functions 339
102. Legendre's Equation . . . . 342
103. Numerical Solution of Differential Equations 346
CHAPTER VIII PARTIAL DIFFERENTIAL EQUATIONS
104. Preliminary Remarks . . .... ... 350
105. Elimination of Arbitrary Functions . . . . . . 351
x CONTENTS
SECTION PAGE
106. Integration of Partial Differential Equations. . . 353
107. Linear Partial Differential Equations with Constant Coefficients . 357
108. Transverse Vibration of Elastic String . . . 361
109. Fourier Series Solution ... . 364
110. Heat Conduction . 367
111. Steady Heat Flow ..... 369
112. Variable Heat Flow ... . . . 373
113. Vibration of a Membrane . 377
114. Laplace's Equation . . ... .... 382
115. Flow of Electricity in a Cable . . 386
CHAPTER IX VECTOR ANALYSIS
116. Scalars and Vectors . ... 392
117. Addition and Subtraction of Vectors 393
118. Decomposition of Vectors. Base Vectors . 396
119. Multiplication of Vectors . . 399
120. Relations between Scalar and Vector Products 402
121. Applications of Scalar and Vector Products . . . 404
122. Differential Operators . . . 406
123. Vector Fields . . 409
124. Divergence of a Vector . 411
125. Divergence Theorem . . . . . 415
126. Curl of a Vector . 418
127. Stokcs's Theorem . 421
128. Two Important Theorems 422 129 Physical Interpretation of Divergence and Curl 423
130. Equation of Heat Flow 425
131. Equations of Hydrodynamics 428
132. Curvilinear Coordinates . . 433
CHAPTER X COMPLEX VARIABLE
133. Complex Numbers . ... . . 440
134. Elementary Functions of a Complex Variable . . 444
135. Properties of Functions of a Complex Variable 448
136. Integration of Complex Functions .... ... 453
137. Cauchy's Integral Theorem . ... 455
138. Extension of Cauchy's Theorem . .... 455
139. The Fundamental Theorem of Integral Calculus . 457
140. Cauchy's Integral Formula . . . . . 461
141. Taylor's Expansion. . . . ... . ... 464
142. Conformal Mapping .... . . . . 465
143. Method of Conjugate Functions . . 467
144. Problems Solvable by Conjugate Functions 470
145. Examples of Conformal Maps . ... ... 471
146 Applications of Conformal Representation . . . 479
CONTENTS xi
SECTION PAGE
CHAPTER XI
PROBABILITY
147. Fundamental Notions . .... 492
148. Independent Events . . 495
149. Mutually Exclusive Events . 497
150. Expectation. . . . 500
151. Repeated and Independent Trials . . . 501
152. Distribution Curve 504
153. Stirling's Formula . 508
154. Probability of the Most Probable Number . . .511
155. Approximations to Binomial Law . 512
156. The Error Function ... 516
157.* Precision Constant. Probable Error . .521
CHAPTER XII
EMPIRICAL FORMULAS AND CURVE FITTING
158. Graphical Method . . 525
159. Differences 527
160. Equations That Represent Special Types of Data 528
161. Constants Determined by Method of Averages 534
162. Method of Least Squares . 536
163. Method of Moments 544
164. Harmonic Analysis . . ... 545
165. Interpolation Formulas . . . 550
166. Lagrange's Interpolation Formula . 552
167. Numerical Integration 554
168. A More General Formula 558
ANSWERS . . 561
INDEX . . 575
HIGHER MATHEMATICS
FOR ENGINEERS AND
PHYSICISTS
CHAPTER I INFINITE SERIES
It is difficult to conceive of a single mathematical topic that occupies a more prominent place in applied mathematics than the subject of infinite series. Students of applied sciences meet infinite series in most of the formulas they use, and it is quite essential' that they acquire an intelligent understanding of the concepts underlying the subject.
The first section of this chapter is intended to bring into sharper focus some of the basic (and hence more difficult) notions with which the reader became acquainted in the first course in calculus. It is followed by ten sections that are devoted to a treatment of the algebra and calculus of series and that represent the minimum theoretical background necessary for an intelligent use of series. Some of the practical uses of infinite series are indicated briefly in the remainder of the chapter and more fully in Chaps. II, VII, and VIII.
1. Fundamental Concepts. Familiarity with the concepts discussed in thig section is essential to an understanding of the contents of this chapter.
FUNCTION. The variable y is said to be a function of the variable x if to every value of x under consideration there corresponds at least one value of y.
If x is the variable to which values are assigned at will, then it is called the independent variable. If the values of the variable y are determined by the assignment of values to the independent
1
2 MATHEM&TICS FOR ENGINEERS AND PHYSICISTS §1
variable x, then y is called the dependent variable. The functional dependence of y upon x is usually denoted by the equation*
V = /(*)•
Unless a statement to the contrary is made, it will be supposed in this book that the variable x is permitted to assume real values only and that the corresponding values of y are also real. In this event the function f(x) is called a real function of the real variable x. It will be observed that
(1-1) y =
does not represent a real function of x for all real values of x, for the values of y become imaginary if x is negative. In order that the symbol f(x) define a real function of x, it may be necessary to restrict the range of values that x may assume. Thus, (1-1) defines a real function of x only if x ^ 0. On the other hand, y — \/x2 — 1 defines a real function of x only if \x > I.
SEQUENCES AND LIMITS. Let some process of construction yield a succession of values
Xij 2*2, #3, , XH) ,
where it is assumed that every xt is followed by other terms. Such a succession of terms is called an infinite sequence. Exam- ples of sequences are
(a) 1, 2, 3, • • • , n, • • • , ,,,1 11 1 / iw i !
(6) 2' - 41 8' ~ 16' ' ' ' '<-*> V (c) 0,2,0, 2, ••-, ! + (-!)", • • • .
Sequences will be considered here only in connection with the theorems on infinite series, t and for this purpose it is necessary to have a definition of the limit of a sequence.
DEFINITION. The sequence x\y x%, • • • , xn, • • • is said to converge to the constant L as a limit if for any preassigned positive number «, however small, one can find a positive integer p such that
\xn — I/I < e for all n > p.
* Other letters are often used. In particular, if more than one function enters into the discussion, the functions may be denoted by /i(x), ft(x), etc.; by/(aO, g(x), etc.; by F(x)t G(x), etc.
t For a somewhat more extensive treatment, see I. S, §okolnikoff? Advanced Calculus, pp. 3-21 f
§1 INFINITE SERIES 5
which is convergent to the value 2. In order to establish this fact, note that
is a geometric progression of ratio J^, so that
J_
on 7~ ^ TvHTZTi"*
Heftce, the absolute value of the difference between 2 and sn is l/2n~1, which can bo made arbitrarily small by choosing n sufficiently large.
On the other hand, if x — — 1, the series (1-4) becomes
which does not converge; for s2n = 0 and S2n~i = 1 for any choice of n and, therefore, lim sn does not exist. Moreover, if x = 2,
n— » oo
the series (1-4) becomes
1 + 2 + 4 + • • • + 2-1 + • • • , so that sn increases indefinitely with n and lim sn does not exist.
n — » oo
If an infinite, series does not converge for a certain value of x, it is said to diverge or be divergent for that value of x. It will be shown later that the series (1-4) is convergent for — I < x < 1 and divergent for all other values of x.
The definition of the limit, as given above, assumes that the value of the limit $ is known. Frequently it is possible to infer the existence of S without actually knowing its value. The following example will serve to illustrate this point.
Example. Consider the series
and compare the sum of its first n terms
_!,!,!, , JL
Sn - l + 2! + 3! + ' ' ' + n!
with the sum of the geometrical progression
6 MATHEMATICS FOR ENGINEERS AND PHYSICISTS
S. = 1 + \ + ± + • • • + ~
The corresponding terms of Sn are never less than those of $„; but, no matter how large n be taken, Sn is less than 2. Consequently, s« < 2; and since the successive values of sn form an increasing sequence of numbers, the sum of the first series must be greater than 1 and less than or equal to 2. A geometrical interpretation of this statement may help to fix the idea. If the successive values of s»,
Si = 1,
«2 = 1 + 21 = 1.5,
ss - 1 + 5j 4- ^ = 1.667,
*4=*l + || + ^ + jj = 1.708,
s* = 1 + I + |j + jj + ^ = 1.717,
are plotted as points on a straight line (Fig. 1), the points representing the sequence Si, $2, * • • , sn, • • - always move to the right but never
0 ,, 1 15 1667 2
* - FIG. 1.
progress as far as the point 2. It is intuitively clear that there must be some point s, either lying to the left of 2 or else coinciding with it, which the numbers sn approach as a limit. In this case the numerical value of the limit has not been ascertained, but its existence was established with the aid of what is known as the fundamental principle.
Stated in precise form the principle reads as follows: // an infinite set of numbers si, §2, * • * , sn, • • • forms an increasing sequence (that is, SN > Sn, when N > n) and is such that every sn is less than some fixed number M (that is, sn < M for all values of n\ then sn approaches a limit s that is not greater than M (that is, lim sn = s < M). The formulation
n— » oo
of the principle for a decreasing sequence of numbers «i, s2, • • * , sn, • • • , which are always greater than a certain fixed number w, will be left to the reader.
2. Series of Constants. The definition of the convergence of a series of functions evidently depends on a study of the behavior
§2 INFINITE SERIES 7
of series of constants. The reader has had some acquaintance with such series in his earlier study of mathematics, but it seems desirable to provide a summary of some essential theorems that will be needed later in this chapter. The following important theorem gives the necessary and sufficient condition for the convergence of an infinite series of constants:
00
THEOREM. The infinite series of constants £ un converges if
n = l
and only if there exists a positive integer n such that for all positive integral values of p
\Sn+p — Sn\ SS \Un+l + Un+2 + ' ' ' + Un+f>\ < €,
where e is any preassigned positive constant.
The necessity of the condition can be proved immediately by recalling the definition of convergence. Thus, assume that the series converges, and let its sum be Sy so that
lim sn = S
n— > oo
and also, for any fixed value of p,
lim sniP = S.
n— > oo
Hence,
lim (sn+p — sn) = lim (un+} + un+z + • • • + un+p) = 0,
n— * « n— * *
which is another way of saying that
for a sufficiently large value of n.
The proof of the sufficiency of the condition requires a fair degree of mathematical maturity and will not be given here.*
This theorem is of great theoretical importance in a variety of investigations, but it is seldom used in any practical problem requiring the testing of a given series. A number of tests for convergence, applicable to special types of series, will be given in the following sections.
It may be remarked that a sufficient condition that a series diverge is that the terms un do not approach zero as a limit when n increases indefinitely. Thus the necessary condition for con- vergence of a series is that lim un = 0, but this condition is not
n— * «o
* See SOKOLNIKOFF, I. S., Advanced Calculus, pp. 11-13.
8 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §2 sufficient; that is, there are series for which lim un = 0 but which
n—> «
are not convergent. A classical example illustrating this case is the harmonic series
in which Sn increases without limit as n increases.
Despite the fact that a proof of the divergence of the harmonic series is given in every good course in elementary calculus, it will be recalled here because of its importance in subsequent con- siderations. Since
_1_ + _J_+ . . • ! ^ ! !
i "i I l r> l
1 n + n " " 2n 2'
it is possible, beginning with any term of the series, to add a definite number of terms and obtain a sum greater than V£. If n = 2,
3 + 4 > 2;
n =' 4,
n = 8,
1, , J_ , , J_ 1.
9 "*" 10 + ' ' ' + 16 > 2; n = 16,
_1_J_ .±^1
17 + 18 + ' ' ' + 32 > 2
Thus it is possible to group the terms of the harmonic series
in such a way that the sum of the terms in each parenthesis exceeds ££,; and, since the series
1+2: + ! + 2:+ '•• is obviously divergent, the harmonic series is divergent also.
§8 INFINITE SERIES 9
3. Series of Positive Terms. This section is concerned with series of the type
an = a! + az + ' ' ' an • • * ,
1
where the an are positive constants. It is evident from the definition of convergence and from the fundamental principle (see Sec. 1) that the convergence of a series of positive constants will be established if it is possible to demonstrate that the partial sums sn remain bounded. This means that there exists some positive number M such that sn < M for all values of n. The proof of the following important test is based on such a demonstration.
oo
COMPARISON TEST. Let 2 an be a series of positive terms,
n = l
00
and let 2 bn be a series of positive terms that is known to converge.
n = l
00
Then the series 2 an is convergent if there exists an integer p such
n = l
00
that, for n > p, an ^ bn. On the other hand, if 2 cn is a series of
n-l
positive terms that is known to be divergent and if an ^ cn for
oo
n > p, then 2 #n is divergent also.
n = l
Since the convergence or , divergence of a series evidently is not affected by the addition or subtraction of a finite number of terms, the proof will be given on the assumption that p = 1. Let sn = «i + a2
00
+ • - • + an, and let B denote the sum of the series 2 bn and Bn its
n = l
nth partial sum. Then, since a» ^ bn for all values of n, it follows that sn ^ Bn for all values of n. Hence, the sn remain bounded, and the
oo
series 2 dn is convergent. On the other hand, if an ^ cn for all values of
n=»l
00 00
n and if the series 2 cn diverges, then the series 2 an will diverge also.
n=l n=l
There are two series that are frequently used as series for comparison.
a. The geometric series
(3-1) a + ar + ar2 + • • • + ar» + • • • ,
10 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §3
which the reader will recall* is convergent to . _ if \r\ < 1
and is divergent if \r\ > 1. 6. The p series
(3-2) l+g+£+ •••+£+•••,
which converges if p > 1 and diverges if p < 1.
Consider first the case when p > 1, and write (3-2) in the form
(3-3)
4-
T
(2n _
where the nth term of (3-3) contains 2n~1 terms of the series (3-2). Each term, after the first, of (3-3) is less than the corre- sponding term of the series
1 J_ O . Jl __ L A . ___ __ L . . . 4_ On— 1
1 ^ ^ ^ ^ ^
2? ^ 4p (2n~1)p '
or
(3-4) 1 + Tj^Zi + /2P-1)2 + ' ' ' + /2p-l)n-l + ' ' ' *
Since the geometric series (3-4) has a ratio l/2p~l (which is less than unity for p > 1), it is convergent and, by the comparison test, (3-2) will converge also.
If p = 1, (3-2) becomes the harmonic series which has been shown to be divergent.
If p < 1, l/np > l/n for n > 1, so that each term of (3-2), after the first, is greater than the corresponding term of the harmonic series; hence, the series (3-2) is divergent also.
Example 1. Test the series
1 + I + I+ ... +!+-..
A T 22 T 33 T -r nn -r
The geometric series
1,1,1, _• 1 .
1 + 22 + 23 + * ' ' + 2* + ' " '
* Since the sum of the geometric progression of n terms a -f or -f or2
i i -i • , . a — arn a X1 N
4- • • • H- arn r is equal to ^_ — = __ (1 — rn).
§3 INFINITE SERIES 11
is known to be convergent, and the terms of the geometric series are never less than the corresponding terms of the given series. Hence, the given series is convergent. Example 2. Test the series
iog~2 loi~3 log! ' k^ •
Compare the terms of this series with the terms of the p series for
given series is divergent, for its terms (after the first) are greater than the corresponding terms of the p series, which diverges when p = 1.
00
RATIO TEST. The series 2 «n of positive terms is convergent if
,. U'n-f-l ^ +
lim — — = r < 1
n— * °° an
and divergent if
lim ^^ > 1.
n— » QO dn
// lim -^ = 1, the series may converge or diverge.
Consider first the case when r < 1, and let q denote some constant between r and 1. Then there will be some positive integer N such that
— < q for all n £ N.
Hence.
aNq, and
Since q < 1, the series in the right-hand member is convergent'; there- fore, the series in the left-hand member converges, also. It follows that
00
the series S ctn is convergent*.
n-l
-
If the limit of the ratio is greater than 1, then an+i > an for every
00
n ^ N so that lim an j& 0, and hence the series S an is divergent.
n— » «o n = 1
12 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §3
It is important to observe that this theorem makes no reference to the magnitude of the ratio of an+\/an but deals solely with tLr limit of the ratio. Thus, in the case of the harmonic series the ratio is an+\/an = n/(n + 1), which remains less than 1 for all finite values of n, but the limit of the ratio is precisely equal to 1. Hence the test gives no information in this case.
Example 1. For the series
and, therefore, the series converges. Example 2. The series
1+1L + 1L+ . . . +"j*L+ .
10 102 103 10"
is divergent, for t. •
r an+i ,. (n + l)!10rt , n lim -^i- = lim v , ' ' -- r- = lim
-- , -- r- — ~r
an n-^oo 10n+1 n! n-K» 10
Example 3. Test the series
Here
r fln+1 "
|
5-6 |
T^ T 7 1 |
2n - l)2n ' . (2n - l)2n |
|
|
4n2 |
- l)(2n + 2) - 2n |
lim ^ X "" ^ |
|
|
4n' + |
6n + 2 , |
^"l +1-4 2n |
1 "2n2 |
Hence, the test fails; but if the given series be .compared with the p series for p = 2, it is seen to be convergent* J*-
oo *
CAUCHY'S INTEGRAL TEST. Let 2 an be a series of positive
n = l
terms such that an+i < an. If there exists a positive decreasing function f(x), for x > 1, such that f(n) = an, then the given series converges if the integral
exists; the series diverges if the integral does not exist.
INFINITE SERIES
13
The proof of this test is deduced easily from the following graphical considerations. Each term an of the series may be thought of as representing the area of a rectangle of base unity and height /(n) (see Fig. 2). The sum of the areas of the first n inscribed rectangles is less than f"+1f(x) dx, so that
f(x) dx.
But f(x) is positive, and hence l f(x) dx
f(x) dx.
If the integral on the right exists, it follows that the partial sums are bounded and, therefore, the series converges (see Sec. 1).
T»vC
The sum of the areas of the circumscribed rectangles, a\ + a 2 + • • • + an, is greater than f "+1 f(x) dx] hence, the series will diverge if the integral does not exist.
Example 1. Test the harmonic series
In this case, f(x) — - and x
f °° 1 Cn dx
I - dx = lim I — = lim log n
Jl X n-» oo Jl X n-> oo
and the series is divergent.
14 MATHEMATICS FOR EN&NEERS AND PHYSICISTS §3
00 i
Example 2. Apply Cauchy's test to the p series ]5£ — - where p > 0.
^^ p
Taking f(x) = — i observe that
-p
l xf l_p- ,. "p*1-
= log a;|?, if p = 1.
/« ^XJ. — exists if p > 1 and does not exist if p < 1. or*
PROBLEMS
1. Test for convergence
l
— __
2 2 • 22 3 • 23 4 • 24
-
/ ^ i i 2! , 3! , (c) 1 + 2i + 35 +
1 ,2,3,
^ + v + 2"3 + • • • ;
21^2 3l^g~3 4Togl "
2. Use Cauchy's integral test to investigate the convergence of
' --
v1'/' A ~ i _j_ 22 ~ 2 + 32
00
3. Show that the series 2 an of positive terms is divergent if nan has a limit L which is different from 0. Hint: Let lim nan = L so that
nan > L — c for n large enough. Hence, an >
n— » oo
- —
§4 INFINITE SERIES 15
4. Test for convergence
1
(2n + I)2' n
4. Alternating Series. A series whose terms are alternately positive and negative is called an alternating series. There is a simple test, due to Leibnitz, that establishes the convergence of many of these series.
TEST FOR AN ALTERNATING SERIES. // the alternating series a\ — c&2 + «a — a* + * * • , where the at are positive, is such that an+i < an and lim an = 0, then the series is convergent.
n— > oo
Moreover, if S is the sum of the series, the numerical value of the difference between S and the nth partial sum is less than an+i. Since
S2n = (ai - a2) + (as — flu) + • • • + («2n-i - a2n) = ai — (a2 — a3) — • • • — (a2n-2 — a2n-i) — «2n,
it is evident that $2n is positive and also that $2n < #1 for all values of n. Also, $2 < $4 < *e < * * • , so that these partial
16 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §6
sums tend to a limit S (by the fundamental principle). Since «2n+i = «2n + «2n+i and lim a2n+i = 0, it follows that the partial
n— » oo
sums of odd order tend to this same limit. Therefore, the series converges. The proof of the second statement of the test will be left as an exercise for the reader.
Example 1. The series
is convergent since lim - = 0 and — — - < -• Moreover. 54
n-> * n n + 1 n
— % + H — y± differs from the sum S by less than y*>. Example 2. The series
--..-.-...
2 "*" 31 4 "*" 32 6 "*" 33 is divergent. Why?
6. Series of Positive and Negative Terms. The alternating series and the series of positive constants are special types of the general series of constants in which the terms can be either posi- tive or negative.
DEFINITION. // u\ + u<t + • • • + un + • • • is an infinite series of terms such that the series of the absolute values of its terms, \Ui\ + \u%\ + • • ' + \un\ + • • • , is convergent, then the series Ui -f- u<i + • • • + un + • • • is said to be absolutely convergent. If the series of absolute values is not convergent, but the given series is convergent, then the given series is said to be conditionally convergent.
Thus,
l-i+i_!+i ____ 2+3 4+5
is convergent, but the series of absolute values,
is not, so that the original series is conditionally convergent.
If a series is absolutely convergent, it can be shown that the series formed by changing the signs of any of the terms is also a convergent series. This is an immediate result of the following theorem:
§6 INFINITE SERIES 17
THEOREM. // the series of absolute values 2 \un\ is convergent,
00
then the series 2 un is necessarily convergent. Let
and
If pn denotes the sum of the positive terms occurring in sn and — qn denotes the sum of the negative terms, then
(5-1) Sn = Pn - qn
and
tn = Pn + qn>
00
The series 2 \un\ is assumed to be convergent, so that
n = l
(5-2) lim tn = lim (pn -f qn) ss L.
n—» oo n— » QO
But pn and gn are positive and increasing with n and, since (5-2) shows that both remain less than L, it follows from the fundamental principle that both the pn and qn sequences converge. If
lim pn — P and lim qn = Q,
n — > * H — > oo
then (5-1) gives
« lim srl = lim (pn — qn) = P — Q,
00
which establishes the convergence of 2 un.
Moreover, it can be shown that changing the order of the terms in an absolutely convergent series gives a series which is convergent to the same value as the original series. * However, conditionally convergent series do not possess this property. In fact, by suitably rearranging the order of the terms of a condi- tionally convergent series, the resulting series can be made to converge to any desired value. For example, it is knownf that the sum of the series
111 f_n»-i
i—i-i-l — ±j_
1 0<0 A 1
-.--
23 4 n
* See SOKOLNIKOFF, I. S., Advanced Calculus, pp. 240-241. t See Example 1, Sec. 13.
18 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §5
p
is log« 2. The fact that the sum of this series is less than 1 and greater than % can be made evident by writing the series as
which shows that the value of sn > % for n > 2; whereas, by writing it as
-»• * O Of \ A f
it is clear that sn < I for n > 2. Some questions might < be raised concerning the legitimacy of introducing parentheses in a convergent infinite series. The fact that the associative law holds unrestrictedly for convergent infinite series can be estab- lished easily directly from the definition of the sum of the infinite series. It will be shown* next that it is possible to rearrange the series
so as to obtain a new series whose sum is equal to 1. The positive terms of this series in their original order are
3' 5' 7' 9'
The negative terms are
11
'
g
In order to form a series that converges to 1, first pick out, in order, as many positive terms as are needed to make their sum equal to or just greater than 1, then pick out just enough negative terms so that the sum of all terms so far chosen will be just less than 1, then more positive terms until the sum is just greater than 1, etc. Thus, the partial sums will be
*2 - - 2 = 2'
- 1,1,1 31 S4==1-2 + 3 + 5 = 30' * General proof can be constructed along the lines of this example.
§8 INFINITE SERIES 19
l * 1 1 47
. . 2 3 5-3 7 9- 1260
1,1,11, 1,1_1_ 1093 1 2 + 3 + 5 4 + 7 "*" 9 6 ~ 1260'
It is clear that the series formed by this method will have a sum equal to 1.
As another example, consider the conditionally convergent series
(5-3) l--4= + -^--4-+---.
' V2 -s/3 VI
Let the order of the terms in (5-3) be rearranged to give the series
LL 4. -1 L\
The nth term of (5-4) is 1
which is greater than
j == •*• _|_ *__ — [i — . \ •*• ,
oo
But the series S &n is divergent, and it follows that the series
n = l
(5-4) must diverge.
00
Inasmuch as the series S |wn| is a series of positive terms, the
n»l
tests that were developed in Sec. 3 can be applied in establishing
00
the absolute convergence of the series 2 un. In particular, the
n = l
ratio test can be restated in the following form:
20 MATHEMATICS FOR ENGINEERS AND PHYSICISTS §5
00
RATIO TEST. The series S un is absolutely convergent if
and is divergent if
lim
lim
Un+l
Un
< 1
> 1.
// the limit is unity, the test gives no information. Example 1. In the case of the series
. z2 . xs . x*
lim
xn (n — 1)!
ft! xn~l
0
for all values of x. Hence, the series is convergent for all values of x and, in particular, the series
22 23
1 - 2 + 21 - 3J + ' ' ' is absolutely convergent. Example 2. Consider