Eroxl's Notes
Geometric Series

A geometric series is any series that follows a geometric sequence.

Vergence

To determine if a geometric series converges typically the ratio test is used.

Ratio Test

The comparison test is a method for determining whether or not a series which can be modelled by a geometric series for large values converges.

Formally, it can be defined as:

Given a series defined in terms of a infinite sequence , given , if the series converges, alternatively if then the series diverges. Additionally, if or the limit does not exist then the test is inconclusive and another test must be used.

Evaluation

A geometric series can be evaluated to a value using the following equation

Where is the common ratio of the sequence that the series follows, is the number the series starts at and is the number the series ends at.

Examples

Non-Infinite Series

The common ratio is and the starting and ending values are and respectively.

Infinite Series

The common ratio is and the starting and ending values are and because the ending value is we need to take the limit as the top terms exponent approaches .