Does a geometric series always have a finite value?

Verified step by step guidance

1. Understand what a geometric series is: it is a series of the form \(\sum_{n=0}^{\infty} ar^n\), where \(a\) is the first term and \(r\) is the common ratio between terms.
2. Recall the condition for convergence of an infinite geometric series: the series converges only if the absolute value of the common ratio satisfies \(|r| < 1\).
3. If \(|r| < 1\), the sum of the infinite geometric series can be expressed by the formula \(S = \frac{a}{1 - r}\), which is a finite value.
4. If \(|r| \geq 1\), the terms do not approach zero, and the series does not converge to a finite sum; instead, it diverges or grows without bound.
5. Therefore, a geometric series does not always have a finite value; it depends on the value of the common ratio \(r\) and whether the series converges.

Key Concepts

Geometric Series Definition

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. It is expressed as S = a + ar + ar² + ar³ + ..., where 'a' is the first term and 'r' is the common ratio.

Convergence of Infinite Series

An infinite series converges if the sum approaches a finite limit as the number of terms increases indefinitely. For a geometric series, convergence depends on the absolute value of the common ratio being less than one (|r| < 1). Otherwise, the series diverges and does not have a finite sum.

Sum Formula for Convergent Geometric Series

When a geometric series converges (|r| < 1), its sum can be calculated using the formula S = a / (1 - r). This formula provides the finite value of the infinite series, illustrating that not all geometric series have finite sums, only those with ratios within the convergence criteria.