Convergence and Divergence of Geometric Sequences

For the geometric sequence , the behavior of the sequence depends on the value of the common ratio :

  1. If , then (diverges).
  2. If , then (converges).
  3. If , then (converges).
  4. If , the sequence oscillates (diverges).
Convergence Conditions for a Geometric Sequence
  1. The necessary and sufficient condition for the geometric sequence to converge is:
  2. The necessary and sufficient condition for the geometric sequence to converge is: