TRUE. The sum to infinity exists if and only if the absolute value of the common ratio is less than 1, written as \(|r| < 1\) or equivalently \(-1 < r < 1\). Why? When \(|r| < 1\), each successive term becomes smaller in magnitude. As n approaches infinity, \(r^n\) approaches 0, causing the terms to approach zero. This allows the sum to converge to a finite value. Examples: r = 0.5: Terms decrease (8, 4, 2, 1, 0.5, ...), sum converges to finite value. r = -0.5: Terms alternate and decrease (8, -4, 2, -1, 0.5, ...), sum still converges. r = 2: Terms increase (2, 4, 8, 16, ...), sum approaches infinity (diverges). r = -2: Terms alternate and grow (2, -4, 8, -16, ...), sum diverges. The formula \(S_\infty = \frac{a}{1-r}\) is only valid when this condition is met. ECZ exams, particularly 2016-2018 papers, frequently test this concept. You must always check \(|r| < 1\) before calculating sum to infinity, or state that it doesn't exist if \(|r| \geq 1\).

Break down complex topics into smaller, manageable chunks for easier recall.

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• Joseph Soko – 101 pts

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