If (constant), for
then
is called a Geometric Sequence or Geometric Progression (G.P). and the series
is called a geometric series. The constant
is known as the common ratio
.
Note -:
(i) If , common ratio
, then
(ii) No term of a GP can be zero, for otherwise will be meaningless for the corresponding value of
.
Partial Sum of a Geometric Series -:
For a geometric series with and common ratio ratio
,
, for
If , then
, for every
, so that
Sum of Geometric Series -:
If <
, i.e.
<
<
then
. When
. So for the geometric series with
<
.
We have
Therefore ; if
<
and
diverges if
.
Hence the geometric series converges if
<
and diverges if
.