Infinite Series
Infinite series are one of the grand themes of analysis. The most spectacular applications (Fourier series, power series, orthogonal series, …) have to do with series of functions; this brief chapter touches only on the underlying fundamentals of series o
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Infinite Series
§10.1. §10.2. §10.3. §10.4.
Infinite series: convergence, divergence Algebra of convergence Positive-term series Absolute convergence
Infinite series are one of the grand themes of analysis. The most spectacular applications (Fourier series, power series, orthogonal series, ... ) have to do with series of functions; this brief chapter touches only on the underlying fundamentals of series of constants. If (sn) is a convergent sequence ofreal numbers, the differences Sn -Sn-l tell us something about the speed of convergence. Writing an = Sn - Sn-l (with the convention So = 0 ), the Sn can be recovered from the an via a telescoping sum Sn = L~=l ak ,thus Sn ----f S means L~=l ak ----f S as n ----f 00 and it is natural to write 00
s= Lak. k=l
The theory of infinite series is the study of such 'infinite sums'.
10.1. Infinite Series: Convergence, Divergence 10.1.1. Definition. If (an) is a sequence of real numbers, the symbol
n=l (or L~ an, or simply Lan) is called an infinite series (briefly, series); an is called the n'th term of the series. We write
179
S. K. Berberian, A First Course in Real Analysis © Springer Science+Business Media New York 1994
10. Infinite Series
180
called the n'th partial sum of the series. One also writes 00
2:= an = al + a2 + a3 + ... . n=l
(This is an equality of symbols- they are equal by definition- not necessarily interpretable as an equality of numbers.) 10.1.2. Example. If e E lR, C:f:. 1, and if an = en - 1 (n = 1,2,3, ... ), then 2 n-l 1 - en sn = 1 + e + e + ... + e = --- .
1-e
10.1.3. Definition. With notations as in 10.1.1, the series is said to be convergent if the sequence (sn) of partial sums is convergent in lR ; if Sn -> s then s is called the sum of the series, we write 00
2:= an =
S,
n=l
and we say that the series converges to s. A series that is not convergent is said to be divergent. Thus, for a convergent series, the symbol L~ an is assigned a numerical value; for a divergent series, we regard it just as a symbol. 1 10.1.4. Example. If lei < 1 then the series
2:= e 00
n -
1
= 1 + e + e2
+ e3 + ...
n=l
is convergent, with sum l~C (see the formula in 10.1.2). If e = 1 then Sn = n and the series is divergent; if e = -1 then Sn alternates between 1 and 0, so the series is again divergent; and if lei > 1 then (sn) is unbounded, therefore divergent, so the series is again divergent. Briefly, the series is convergent when lei < 1 , divergent when lei::::: 1 . It is called a geometric series with ratio e (each term is obtained from its predecessor by multiplying by c). 10.1.5. Remarks. Let C be a condition that an infinite series Lan mayor may not satisfy. If C true
=?
2:= an convergent,
1 Just the same, there are various devious ways of assigning numerical values to the symbol even for certain divergent series (the theory of 'summability'). Cf. Exercise 2.
§10.1. Convergence, Divergence
181
then C is called a sufficient condition for convergence. If
L an convergent
C true,
=}
then C is called a necessary condition for conver
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