What is geometric series?

A geometric series is a series with a constant ratio between successive terms. This ratio is called the common ratio.

Form: A geometric series can be represented in the form: a + ar + ar² + ar³ + ... where:

• a is the first term
• r is the common ratio

Sum of a Finite Geometric Series:

The sum (S_n) of the first n terms of a geometric series is given by the formula:

S_n = a(1 - rⁿ) / (1 - r), if r ≠ 1

Sum of an Infinite Geometric Series:

An infinite geometric series converges (has a finite sum) only if the absolute value of the common ratio is less than 1 (|r| < 1). The sum (S) of a convergent infinite geometric series is:

S = a / (1 - r), if |r| < 1

Convergence and Divergence:

• If |r| < 1, the geometric series converges
• If |r| ≥ 1, the geometric series diverges (unless a = 0, in which case the sum is trivially 0).

Applications: Geometric series have applications in various fields, including:

• Finance (e.g., calculating compound interest)
• Physics (e.g., modeling radioactive decay)
• Computer science
• Probability