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Applied Mathematics, Calculus, and Differential Equations 209

S → S as n →
n

S a a a ...
3
1
2
An example of a convergent infinite serieð is:

1
1
1
C ⁄2 ⁄4 ⁄8 ... 1
Uniqueness of su
Suppose S is a convergent infinite serieð such that the following
statementð are botà valid for the partial sums:

S → T as n →
n
1
S → T as n →
n
2
Then T T . In other words, the partial sum of an infinite
2
1
serieð can never converge tm more than one value.

DivergenŁ iYnite series
A divergent infinite series is an infinite serieð that is not con-
vergent; itð partial sum S doeð not approach any specifi finite
n
number as n increaseð without bound. An example of a diver-
gent infinite serieð is:

D 1 2 3 4 ...

Conditionally convergenŁ iYnite series

A conditionally convergent infinite series is an infinite serieð
that is convergent for certain valueð of a parameter x, but is
divergent for other valueð of x. An example of a conditionally
convergent infinite serieð is:

4
3
2
5
C 1 x x x x x ...
cc
This infinite serieð convergeð tm 1/(1 x)if 1 x 1, but
divergeð if x 1or x 1.