Geometric Series

A series derived from the geometric sequence , where the first term is and the common ratio is , is called a geometric series. It is expressed as:

Convergence and Divergence of Geometric Series

  1. The partial sum of the geometric series (the sum up to the th term) is given as follows:
    1. When :
    2. When :
  2. The conditions for the convergence of the geometric sequence (where ) are:
    The conditions for the convergence of the geometric series are:

Convergence and Divergence of the Geometric Series

The geometric series (with ) behaves as follows:

  1. Convergence: If , the series converges, and the sum is:
  2. Divergence: If , the series diverges.