Conditional Convergence
A series that converges but does not converge absolutely converges conditionally .
i.e.,
converges conditionally (is conditionally A series
convergent)if the corresponding series of absolute value is convergent. is divergent and
Notation:
be an alternating series then a series Let
is conditionally convergent if
is convergent. is divergent and
Example:
The following is conditionally convergent series:
Solution:
For
the corresponding series of absolute values is:
diverges because it is a harmonic series. The series
with p = 1
To determine the following alternating series
, clearly: converge: Here
, and (1)
. = (2)
Hence we deduce from the alternating series test that the given series converges .
converges . i.e.,
converges conditionally . Therefore
Exercise:
The following are conditionally convergent series:
. (1)
. (2)
. (3)
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