/v\.
i! tt.
STAT.
UBRAIt
INTRODUCTION TO
INFINITE SERIES BY
WILLIAM
F.
OSGOOD,
Pir.D.
ASSISTANT PROFESSOR OF M...
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/v\.
i! tt.
STAT.
UBRAIt
INTRODUCTION TO
INFINITE SERIES BY
WILLIAM
F.
OSGOOD,
Pir.D.
ASSISTANT PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY
CAMBRIDGE b$ Ibarparfc
1897
"Ulniversitp
Copyright, 1897, by
HARVARD UNIVERSITY.
fft fl/ffti
PKEPACE. TN an introductory course on the
Differential
and Integral Calculus
the subject of Infinite Series forms an important topic.
presentation of this subject should have in view
first to
make
The the
beginner acquainted with the nature and use of infinite series and secondly to introduce him to the theory of these series in such a that he sees at each step precisely
what the question
never enters on the proof of a theorem actually requires proof.
till
at issue is
way and
he feels that the theorem
Aids to the attainment of these ends are
(a) a variety of illustrations,
:
taken from the cases that actually arise
in practice, of the application of series to computation both in pure
and applied mathematics (b) a meaning and scope of the more ;
full
diagrams and graphical illustrations
The pamphlet
that follows
the kind here indicated.
The
is
and careful exposition of the
difficult
theorems
;
(c)
the use of
in the proofs.
designed to give a presentation of
references are to Byerly s Differential
Calculus, Integral Calculus, and Problems in Differential Calculus ,
and to B. O. Peirce
Ginn
&
s
Short Table of Integrals;
all
published by
Co., Boston.
WM. CAMBRIDGE, April 1897.
F.
OSGOOD.
LNTKODUCTIOISr.
1.
Consider the successive values of the variable
Example.
=
sn
for
n
=
r
1 -f-
r2
-f-
-f~
rn
~1
Then
Let r have the value J.
1, 2, 3,
s2 *3
If the values
-|-
= =
+J - +4+i 1
i
be represented by points on a S,
=
line, it is
Sa
I
FIG.
1J if
easy to see the
S,
S4
2.
1.
law by which any s n can be obtained from its predecessor, lies half way between * H _ 1 and 2. namely Hence it appears that when n increases without limit, :
The same
result could
formula for the sum
a
-f-
s n of
,
ar
=
1, r
=
s
tl
=.
2.
have been obtained arithmetically from the the first n terms of the geometric series
-\-
ar 2
-\-
a (I -
8
When
l
/t
Lim
Here a
sn _
.s
-\-
ar n-1
,
r)
J,
w increases without
and hence as before Lim
SH
limit,
=. 2.
^ approaches
as
its
limit,
INTRODUCTION.
2
2.
any
an
Definition of
2.
Let M O w 1? w 2
Infinite Series.
,
+
U
Denote the sum of the
Allow n
1
.....
+M + 2
n terms by
first
sn
be
series
(1)
:
Then
to increase without limit.
..... ,
and form the
set of values, positive or negative or both,
either a) sn will
approach
U:
a limit
Lim n or b) s n approaches no limit. Infinite Series, because n
is
=
SH
=
U;
co
In either case we speak of (1) as an allowed to increase without limit. In
case a) the infinite series is said to be convergent and to have the In case b) the infinite or converge towards the value U.
U
value*
series is said to
be divergent. series above considered
The geometric vergent
is
an example of a con
series. 1 1
..... .....
+ 2 + 3+ + 1-1
are examples of divergent series.
,
Only convergent
series are of use
in practice.
The notation uo is
+u +
.....
often used for the limit
U *
d
\
C7,
u
inf. (or to infinity)
or simply
+u +
.....
v
often called the sum of the series. But the student must not forget not a sum, but is the limit of a sum. Similarly the expression "the sum of an infinite number of terms" means the limit of the sum of n of these terms,
that
as
7
is
7 is
n increases without
limit.
I.
a)
3.
ALL OF WHOSE TERMS ARE POSITIVE.
SERIES,
Let
Example.
CONVERGENCE.
it
be required to test the convergence of the
series
where n\ means 1-2 3
....... n
moment
Discarding for the
the
first
and
read
is
factorial
term, compare the
sum
n".
of the
next n terms
19ft IZIZO
1.9
9 l^O
l
with the corresponding
1
ft
sum
2
n ^L
"
1 -I }V)
1
M
|
1
by
r
O^J
|
1 ^>
=
I
T
converges toward the limit
many terms
1
Ap
I
1
I
.
Qp
1
+
.
p
L- 41
1- H J
p
then, since
;
get
^ -f -JL +
+)p
_i_
Denote 1/2
we
0, r
r -f r 2 I
and the
1
series
2^2
3
a convergent series, for
7 p
r 4
3
1.01.
J
7
4
Now
consider what the nu
merical values of these roots in the denominators are
In fact
^ 2 = 1.007, ^ 3 = 1.011, ^ 4 ^ 100 = 1.047 and ^ 1000 = 1.071;
thousand terms of the last
term
is
can
+_l_ + ^_+ + _J_ 9 OA/O 4:y4: zyz
for every value of
is
series,
:
1.014. that
is,
when a
have been taken, the denominator of the a number so slightly different from 1 that multiplied by series
significant figure of the decimal part appears only in the second place. And when one considers that these same relations will be still more strongly marked when p is set equal to 1.001 or 1.0001, one may well ask whether the series obtained by putting p 1,
the
first
1.
is
not also convergent.
1
1
CONVERGENCE.
8
This
is
11 I
r
n+l
since each of the
we can
For
however not the case.
?i
1
r
n terms, save the
last, is
111 ^V.
I
+2n
7, 8.
n
+ n^
f^\
2n~~2
greater than 1/2 n. Hence add a definite number of
strike in in the series anywhere,
terms together and thus get a sum greater than this as often as
we
,
and we can do
For example,
please.
i+\>\
j+i+t+t>j "=
Hence the sum
++
-
the
of
first
+>
n terms increases without
n
limit as
increases without limit,
Lim
or
n
The
series (4) is called the
How
=
sn
=
oo
oo
harmonic
series.
the apparently sudden change from convergence for p 1 to be accounted for? in series (3) to divergence when p is
^>
1
The
When p is only slightly greater than 1, explanation is simple. series (3) indeed converges still, but it converges towards a large value, and this value, which
is
of course a function of p, increases
without limit when p, decreasing, approaches limit exists,
8.
and the
Test for Divergence.
Exercise.
namely
When p
5 for convergence,
Let o
+w +
(
i
of positive terms that is to be tested for divergence. series of positive terms already known to be divergent be
a
no
1,
Establish the test for diver
gence of a series corresponding to the test of :
1.
series is divergent.
series
o
+
i
+
a)
If a
(ft)
can be found whose terms are never greater than the corresponding terms in the series to be tested (a), then (a) is a divergent series.
Examples. ,
.
J_4.
J
.
^
j_
+ ^
9
CONVERGENCE.
8, 9.
1
1
_
i
4_ i
_i_
i
i _L I
JL
i
_L_
This last series can be proved divergent by reference to the series
2
Let
""
4
""
"""
6
111 + + + 246
-
sn
1
+ v~ 2n
-
-
series in parenthesis is the harmonic series, and its sum in hence S H increases without limit creases without limit as n increases
The
;
and the
series is divergent.
The
Second Test for Convergence.
9.
Test- Ratio.
Let the series
to be tested be
+ Ul +
U and form the
When n definite
Then
test-ratio
increases without limit, this ratio will in general approach a fixed limit (or increase without limit). Call the limit r.
if r
the series is convergent,
1
r
"
First, let r
1.
1,
it
is
divergent, if
:
Then
as
=
1,
Convergent ; Divergent;
No
Test.
n increases, the points corresponding about the point r, and hence if
u, + 1 /u n will cluster l
r
o
1
I
FIG.
y 4
i
1
4.
a fixed point y be chosen at pleasure between T and 1, the points u n + l /u n will, for sufficiently large values of n, i.e. for all values of n equal to or greater than a certain fixed number m, of y,
and we
shall have
lie
to the left
CONVERGENCE.
10
m
n =-
Adding p
here the third hypothesis of the theorem comes
hence
;
__ -
Ul
Thus
2
m=oo
QO
TT
over
U
=:
+ 1 z= lim
into play for the first time
or simply U.
2//t
U
limit,
U2
approaches a
J
- --
s rt
its limit.
TT
limit,
o
1
1
llll FIG.
I
C7,
continually springing
vHi^ 1
and
s2
s 2w
sx .
1,
*
>
m.
1(5,
CONVERGENCE.
17.
u
Hence
"
i
that
the
is, all
=
quantity p limit,
if
B
s
uu
or
^w
>
=
>
;
u n and u n cannot approach
as their
oo.
In the series of
Example.
m
n
,
= m on are greater than a certain positive
from
u m and hence
when n
19
=
15,
a;;
hence
verges for all values of x numerically greater than may be represented graphically as follows
1
this series di
These results
.
:
1
1
Divergent
Divergent
Convergent
For what values of x are the following series conver what values divergent? Indicate these values by a diagram
Exercise.
gent, for similar to the one above.
x
x*
"
. |
I
H
V~2
V3
Ans. a;
1
1,
x
1,
t
,
/a rt
respectively, // a and form the M,
positive quantity greater than any of the quantities p series
20
CONVERGENCE.
The terms
17, 18.
of this series are less respectively than the terms of the
convergent series
Ha and each
.....
+ Ha\ + //a + 2
made up exclusively of positive terms. converges and the series
Hence
series is
first series
the
converges absolutely.
Examples.
1
The
.
series
since
sin 3
T
~~^~
,
converges absolutely for
all
I
2
32
converges absolutely and sin If
-f-
!
+
-f- a-!
and
cos x
6 X sin
2
nx
never exceeds unity numerically. an( l
^1
+
+
^2
.....
are
series, the series
+a
-f- 6 2
a?
series
.....
+ I-T 5
any two absolutely convergent a
For the
x.
.....
+
tf
~~5*~
values of
I. -1
2.
sin5#
a;
"
2
2
cos 2 x
SU1 2a; -f-
..... .....
-f-
converge absolutely. 3.
Show
that the series e"*
cos x
-f-
e~ 2;r cos 2x
.....
-\-
converges absolutely for all positive values of x. 4. What can you say about the convergence of the series 1
+
r
+r
cos
2
cos 2
.....
-f
18. Convergence and Divergence of Power Series. ascending integral powers of a;,
where the called a aj,
but
others.
coefficients
power
it
series.
a
will in general
In the
,
a l5 a 2
Such a
,
series
+
series of
j
may converge
for
all
of #,
is
values of
converge for some values and diverge for
latter case the interval
distances in each direction Divergent
A
..... ..... are independent
a o 4~ a i x H~ a x2 (7
J
These integrals can be evaluated by the aid of the formulas of IV of Short Table of Integrals. In particular, the length of a will S be found quadrant by putting ^ TT and using the formula 240 of the (No. Tables)
Peirce
s
=
(Cf Byerly s .
Hence Z
first
N=
denotes the length of the radius of the earth.
Diff. Cal.,
the
primed
true time at (A), then
exceed 5 miles, li/E
(aR D
being continuous near
(#)
35
=
a?
a 1
"
/
(o
b )=|=
0,
.
Show
that a perpendicular drawn to the tangent from a point is an infinitesimal of infinitely near to a point of inflection
2.
P
.....
P
higher order than the second. The osculating circle was defined (Diff. Col. 90) as Curvature. a circle tangent to the given curve at and having its centre on the
P
We
inner normal at a distance p (the radius of curvature) from P. and a be taken infinitely near to will now show that if a point
P
P
perpendicular
FM be dropped from P P
the osculating circle at P", then of the third order referred to the arc
Let
on the tangent at P, cutting in general an infinitesimal
is
P"
PF
as principal infinitesimal.
P
P
be taken as the origin of coordinates, the tangent at being and let the ordinate the axis of x and the inner normal the axis of y Here y be represented by the aid of (13). ;
*
=
0,
x=h, y =
and
The radius
/(0)
$f"(0)x*
of curvature at
P is
and the equation of the osculating *
Hence the
*
=
+
circle is 2
(y
p)
=
P-
lesser ordinate y of this circle is given
y r= p
n
2
V
2
p
*?
-
=
P
P (1
I
* by the formula :
I
Instead of the infinite series, formula (13) might have been used here, with But we happen to know in this case that the function can be developed
4.
by Taylor
s
Theorem
(15).
TAYLOR
40
and
y
y
S
THEOREM.
31, 32.
= )
K*^*>:-*?From
this
result
follows
B
that
y )/x approaches in general a finite limit different from 0, and hence that ?/ y is an infini tesimal of the third order, referred to x as principal infini
tesimal.
PM
But
proposition. Exercise.
Show
(y
PM
and
PP
are of
the
same
order.
that for any other tangent circle y
.
Hence the y
is
an
infinitesimal of the second order.
Second Application : Error of Observation Let x denote the magni tude to be observed, y the f (x) magnitude to be computed from the observation. Then if .r be the true value of the observed magni .
=
tude, x =. x -\- h the value determined by the observation, h will be the error in the observation, and the error // caused thereby in the result will be (cf. (14))
H = f(x + h)
f(x
)
= f (x
+0h)h.
In general / (#), will be a continuous function of x and thus the value of f(x Oh is Oh) will be but slightly changed if a?
+
replaced by x.
+
Hence, approximately,
H = f (x)h and
this is the
formula that gives the error in the result due to the
error in the observation.
The Principal Applications of Taylor s Theorem without the Remainder, i. e. Taylor s Series (15) consist in showing that the 1 fundamental elementary functions e x sin a, cos a-, logo;, o^, SB, 32.
sin"
:
,
1 tan"
a;
can be represented by a Taylor
s
explicitly the coefficients in these series.
Series, It is
and
in
determining shown in Ch. IX of
the Diff. Cal. that these developments are as follows.*
2!
1
xs
z5
,
5!
3! COS
99=1
1
These developments hold for *
The developments
x4
x*
all
values of x.
for sin" a; and tank s are to be sure obtained by in tegration; but the student will have no difficulty in obtaining them directly
from Taylor
s
Theorem.
1
TAYLOR
32, 33.
log*
-i
l tan~"
T
a?
=
log(l
Z
* 4.
"
=
+
1
+ h) =
XS
S
-
h
3
^
2^4
5
1
+
THEOREM.
1-
+
,
;
-J-
o
o
These developments hold for all values of 7i last two formulas, of a?) numerically less than Exercise.
Taylor
s
-
-4-
-I-
2 3
+
41
Show
(or, in the
case of the
1.
that sin a; can be developed about any point X Q by that the series will converge for all values of h.
Theorem and
Hence compute
sin 46
correct to seconds.
33. As soon however as we pass beyond the simple functions and Theorem, we encounter a difficulty that is In order namely to show that f(x) can be usually insurmountable. expanded by Taylor s Theorem it is necessary to investigate the general expression for the n-ih derivative, and this expression is try to apply Taylor s
To avoid this difficulty recourse usually extremely complicated. methods of obtaining the expansion. more or less indirect is had to For example,
let it
be required to evaluate
ra
/ The
indefinite integral
do;.
cannot be obtained and thus we are driven to
develop the integrand into a series and integrate term by term. Now x e~ r )/x^ if we try to apply Taylor s Theorem to the function (e the successive derivatives soon
ever proceed as follows
become complicated.
We
:
= -x+
X1
1
and hence, dividing through by
+
.T,
X - 3T + s
we have
3l
+
5l^
.....
can how
TAYLOR
42
^
dx=2
Examples.
Do
(l
S
THEOREM.
+ 3 37+5^7+
33.
)
made up
functions, into a
502-
the examples on p. 50 of the Problems.
General Method for the Expansion of a Function. function /(a?),
= *.H4
power
To
develop a
manner out of the elementary the general method is the following.
in a simple series,
The fundamental elementary functions having been developed by 32, we proceed to study some of the simplest Taylor s Theorem, operations that can be performed on series and thus, starting with the developments already obtained, pass to the developments de sired.
ALGEBRAIC TRANSFORMATIONS
IV.
OF SERIES. has been pointed out repeatedly ( 19, 21, 24) that since not a sum, but a limit of a sum, processes appli if applicable, this cable to a sum need not be applicable to a series
34.
an
It
infinite series is
;
fact requires proof. For example, the value of a
which the terms are added. terms be extended to series?
value
Its
less
is
terms as follows i
The
than
1
-)-
i
=
(12)
f
+ * - i + i + i- * -M + A -J +
general formula for three successive terms
if
the result it is
is
its
Rearrange
3
4 A;
1
.....
(i
2k
+ I) - i + (i + A)
an alternating
(-8)
is
each pair of positive terms be enclosed in parentheses
+ i) - i +
(i
For
J
:
4k and
sum is independent of the order in Can this interchange in the order of the Let us see. Take the series
*
:
+
(y)
series of the kind considered in
(
11).
easy to verify the inequalities
Hence the
converges toward a value greater than (1 -)- ^) of the first n terms of (/?) differs from a properly chosen sum of terms of (y) at most by the first term of a parenthesis, series (y)
The sum
J z= J.
a quantity that approaches as its limit when n =. Hence the series (/?) and (y) have the same value and the rearrangement of <x>
terms in
from
(a)
has thus led to a series
(/?)
.
having a different value
(a).
In fact
it is
possible to rearrange the terms in (a) so that the
have an arbitrarily preassigned value, C. For, if positive, say 10 000, begin by adding from the positive terms series will
new
C
is
ALGEBRAIC TRANSFORMATIONS OF SERIES.
44
enough have been taken so that
till
their
sum
34, 35.
will just
exceed C.
This will always be possible, since this series of positive terms Then begin with the negative terms diverges.
and add just enough
to reduce the
sum below
As
C.
soon as this
has been done, begin again with the positive terms and add just enough to bring the sum above (7; and so on. The series thus obtained value
is
is
the result of a rearrangement of the terms of (a) and its
C.
In the same
way
it
can be shown generally that
+
"o
"i
+w + 2
if
.....
any convergent series that is not absolutely convergent, its terms can be so rearranged that the new series will converge toward the pre-
is
Because of
assigned value C.
such series are often called
this fact
35 justifying the denoting conditionally convergent, Theorem 1 of of absolutely convergent series as unconditionally convergent. There is nothing paradoxical in this fact, if a correct view of the
For a rearrangement of nature of an infinite series is entertained. terms means a replacement of the original variable s n by a new vari able s n in general unequal to S H and there is no a priori reason why ,
,
these two variables should approach the same limit. The above example illustrates the impossibility of extending a Most of such priori to infinite series processes applicable to sums.
processes are however capable of such extension under proper restric and it is the object of this chapter to study such extension for some of the most fundamental processes.
tions,
35. THEOREM
In an absolutely convergent series the terms can
1.
be rearranged at pleasure without altering the value of the series.
suppose
First, sn
all
the terms to be positive and let
=u + u +
.....
1
+
_!
n
After the rearrangement
U.
lim s n
;
=
co
let
=
For s n always in approaches the limit U when n creases as n increases but no matter how large n be taken (and then held fast), n can (subsequently) be taken so large that s n will include
Then
sn
,
;
all
the terms of
s
>
and more too
;
therefore
<x>
.
.
ALGEBRAIC TRANSFORMATIONS OF SERIES.
35.
45
no matter how large n be taken,
or,
U.
,
7?
^Y!.
Then
le
/(XO-/W i.
65
Cl