/v\. i! tt.
STAT.
UBRAIt
INTRODUCTION
TO
INFINITE SERIES
BY
F.WILLIAM Pir.D.OSGOOD,
PROFESSOR OF MATHEMATICS INASSISTANT HARVARD UNIVERSITY
CAMBRIDGE
Ibarparfcb$ "Ulniversitp
1897Copyright, 1897, by
UNIVERSITY.HARVARDfft
fl/ffti
PKEPACE.
an course on the Differential and CalculusTN Integralintroductory
the of Infinite forms an TheSeriessubject important topic.
of this should have in view first to make thepresentation subject
with the nature and use of infinite series andbeginner acquainted
to introduce him to the of these series in such a waysecondly theory
that he sees at each what the at issue is andstep questionprecisely
never enters on the of a theorem till he feels that the theoremproof
Aids to the attainment of these ends are :actually requires proof.
a of taken from the cases that arisevariety illustrations, actually(a)
in of the of series to both inpractice, application computation pure
and mathematics a full and careful of the
;applied (b) exposition
and of the more difficult theorems the use of
;meaning scope (c)
and illustrations in thediagrams graphical proofs.
The that follows is to a ofpamphlet designed give presentation
the kind here indicated. The references are to sByerly Differential
inand Problems CalculusCalculus, Integral Calculus, ,Differential
and to B. O. Peirce s Short Table allof Integrals; published by
Ginn & Boston.Co.,
F.WM. OSGOOD.
1897.CAMBRIDGE, AprilLNTKODUCTIOISr.
Consider the successive values of the variable1. Example.
~n2 1s = r r r1n -f- -|- -f- -f~
= r the value Thenfor n Let have1, 2, 3, J.
s 1 =
2 + J 1J
* - i =
3 + 4+ i if
If the values be on a it is to see therepresented points line,by easy
=
I S S 2.S, S,a 4
FIG. 1.
law which s can be obtained from its s _any n predecessor, n l ,by
: .s lies half between * _ and 2.namely /t way H 1
Hence it that when n increases withoutappears limit,
=.Lim s 2.
tl
The same result could have been obtained from thearithmetically
s nformula for the sum of the first terms of the seriesn geometric
2 n-1ar ara ar
-f- -\- -\- -\- ,
a
(I r)
-8
= r=Here a 1, J,
When w increases without as itslimit, approaches limit,^
=.and hence as before Lim S 2.H2 INTRODUCTION. 2.
.....an Series. Let M w w be2. Definition of Infinite O , 1? 2 ,
set of or or and form the seriesany values, positive negative both,
.....U M
1 2+ + + (1)
of the first n terms s :Denote the sum nby
Allow n to increase without limit. Then either s willn approacha)
a limit U:
=Lim S
H U;
=n co
or s no limit. In either case we of as an
n approaches speakb) (1)
because n is allowed to increase without limit. InInfinite Series,
case the infinite series is said to be and to have theconvergenta)
value* U or towards the value U. In case the infiniteconverge b)
series is said to be divergent.
is anThe series above considered of a congeometric example
series.vergent
.....
1 2 ,+ + 3+1 1--1 +
are series. series are ofof useexamples convergentdivergent Only
in practice.
The notation
.....u u d to
o+ \+ inf. (or infinity)
is often used for the limit or
C7, simply
.....U u u+ v+
* must not7 is often called the sum of the series. But the student forget
7 the "the sumthat is not a but is the limit of a sum.sum, Similarly expression
the limit of the sum of n of theseof an infinite number of terms" means terms,
as n increases limit.withoutI. CONVERGENCE.
ALL OF WHOSE TERMS ARE POSITIVE.a) SERIES,
Let it be to test the of the3. Example. required convergence
series
.......where n\ means 1-2 3 n and is read n".factorial
for the moment the first the sum of theDiscarding term, compare
next n terms
l1.9 19ft 1 9 ftIZIZO l^O
with the sumcorresponding
2
n 1 factors
(Cf. 1).
^L<2
Each term of a- after the first two is less than then corresponding
in and the sumterm S. hencen ,
the discarded term and the sum of the first nor, inserting denoting
terms of the series Sgiven by M ,
no matter how ?i be taken. That is to s is a variablelarge say, n
nthat increases as but that never attains soalways increases, large
a To makevalue as 3. these relations clear to the theeye, plot
successive values of s as on a line.
n pointsCONVERGENCE. 4.3,
s = 1 1 =2.
2 +
1-* = 1 1 =2.5
3 + +
JLjL$ = 2.667
4 l-fl + +
l = 2.708*5=l + + + +7^j- f7
111 1
~ -9717*6
~2~!~^3~! ~4T ~5l
- I I I I -_i_ _i is j ]_ 2 718
=o s,-
1H
When n increases the s movesby 1, point representing H + 1 always
to the but never advances so far as the 3. Hence thereright, point
must be some either with 3 or to the 3point e, coinciding lying left of
e <C which s as its but never reaches. Ton(i.e. 3), approaches limit,
from the values for s s <s the value of e to2 ,judge computed 1? 8 ,
three of decimals is a fact that will be established2.718,places
later.
4. FUNDAMENTAL PRINCIPLE. The which we havereasoning by
existence of ahere inferred the limit we do not ase, although yet
how the numerical value ofknow to that is ofcompute limit, prime
for the work that follows. Let us state it inimportance clearly
form.
general
s is a variable that increases wlien n increases:If n always1)
but that remains less than some A:always definitefixed number,2)
s A
tt
<
values then s aall U:for of n, n approaches limit,
Lim fs = U.
ltCONVERGENCE.5.4,
is not than A :This limit, U, greater
U<A.
s s s U As, a 3 4
1
1 1 hH+H
be the limit itself or value than theThe valueA any greatermay
limit.
Exercise. State the for a variable that is dePrinciple always
but is than a certain fixed andcreasing, always greater quantity,
draw the corresponding figure.
5. First Test Direct On thefor Convergence. Comparison. prin
of the is based the test for theciple preceding paragraph following
of an infinite series.convergence
.....u uLet a\ z+ + + ( )
the it isbe a series which desired toterms, ofofpositive convergence
a series terms known to betest. ofpositive alreadyIf convergent
can be whose terms are never less than the termsfound corresponding
in the series to-be tested then is a and its value(a), (a) convergent series,
does not exceed that the seriesof ((3).
For let
.....=
n o i <V-i,
Lira S = A.
n
=n QO
Then since S A and s <C Sn
<^ n n ,
it follows that s A
n <^
and hence 4 s a limit and this limit is notby n approaches greater
than A.
Remark. It is convenient in thefrequently studying convergence
of a series to discard a few terms at the whenmbeginning (m, say,
& andis to consider the new series thus Thatfixed number) arising.
of isthe this series and sufficient for theconvergence necessary
of isthe series sinceevident, original6 CONVERGENCE. 6.5,
u is constant and hence s will toward a limit if s _n converge n m does,
and conversely.
Prove the seriesExamples. following convergent.
4 9 16 .....
r r r r r 1+ + + + , < <
- - -+ +^ ^ ^
3 ! 5 ! 7 !
1-2 2-3 3-4
Write s in the fo-inSolution.
+!
then
=Lim s 1
n
1 2 8 4 o-6
, , .1I 1j 4.
1^ T^ 2 I I
22 ^2g
..... 2l+ + + , P
>
A New Test-Series. It has been seen that the series6. just
We will nowwhen the constant > 2.p proveconverges quantity
1 . The truth of thethat it also wheneverp followingconverges >
is at once evident :inequalities
2
_ J_I 1 =+ <-
"^ ^ ~ 1
/)2? 3* 2* 2
JLIII !<-!-
T~ ~r 1 ^^ ,
p P ;>4; 5 Q 7 4/ ^-i
I_~ ~I- i<r^-^P "- 1P P