Theorem 4.2.9: Geometric Series

Let a be any real number. Then the series is called Geometric Series.
  • if | a | < 1 the geometric series converges
  • if | a | 1 the geometric series diverges
If the geometric series converges (i.e. if | a | < 1) then
=
Context Context

Note that the index for the geometric series starts at 0. This is not important for the convergence behavior, but it is important for the resulting limit.

Examples 4.2.10:
 
  • Investigate the convergence behavior of the following series:
    1. What is the actual limit of the sum ?
    2. What is the actual limit of the sum ?
    3. Does the sum converge ? (Here the limit comparison test may be helpful).
Proof:

The proof consists of a nice trick. Consider the partial sum S N and multiply it by a:

Subtracting both equations yields: (1 - a) SN = 1 - a N+1. Dividing both sides by (1 - a) and taking the limit, the result follows from previous result on the power sequence.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007