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 .
(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 ,
If , then , for every , so that
Sum of Geometric Series -:
If < , i.e. < < then . When . So for the geometric series with < .
Therefore ; if < and diverges if .
Hence the geometric series converges if < and diverges if .