Page 217 -

210 Chapter Three

Absolutely convergenŁ iYnite series
An absolutely convergent infinite series is an infinite serieð such
that the sum of the absolute valueð of all itð termð is conver-
gent. Let

S a a a ...
1
2
3
be an infinite series. Then S is absolutely convergent if and only
if the following serieð converges:

S abð a a a ...
1
2
3
Alteration of initial terms in series

Let S and T be infinite serieð that are identical except for the
first k terms; the first k termð have different valueð a and b ,
n n
as follows:

S a a a ... a a a a ,...
1 2 3 k k 1 k 2 k 3
T b b b ... b a k 1 a k 2 a k 3 ,...
k
3
2
1
Then the following statementð are always true: if S is conver-
gent, then T is convergent; and if S is divergent, then T is di-
vergent.

Terms in convergenŁ iYnite series
If S is a convergent infinite series, then the corresponding se-
quence A convergeð toward zero. Let S, a partial sum S , and
n
A be denoted as follows:

S a a a ...
1
2
3
S a a a ... a
n 1 2 3 n
A a , a , a , ...
3
1
2
If S → S as n → , then a → 0as n → , where S is a specific,
n
n
unique real number.