87. Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.

g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval (1/2, ∞).

검증된 단계별 안내

Recall the formula for the sum of an infinite geometric series \(S = 1 + r + r^2 + r^3 + \cdots\) when \(|r| < 1\), which is given by \(S = \frac{1}{1 - r}\).

Identify the domain of \(r\) for which the series converges: the series converges only if \(|r| < 1\).

Analyze the range of the sum \(S = \frac{1}{1 - r}\) as \(r\) varies within \((-1, 1)\): as \(r\) approaches 1 from below, \(S\) grows without bound towards \(+\infty\); as \(r\) approaches \(-1\) from above, \(S\) approaches \(\frac{1}{1 - (-1)} = \frac{1}{2}\).

Conclude that the sum \(S\) takes on all values in the interval \((\frac{1}{2}, \infty)\) as \(r\) varies in \((-1, 1)\), which is the domain of convergence.

Therefore, the statement is true if we consider \(r\) restricted to \((-1, 1)\), but false if \(r\) is allowed outside this interval since the series does not converge there.

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Geometric Series and Its Sum Formula

A geometric series is a sum of terms where each term is a constant ratio r times the previous term. For |r| < 1, the infinite series 1 + r + r² + r³ + ⋯ converges to 1/(1 - r). Understanding this formula is essential to analyze the values the series can take as a function of r.

Domain and Range of the Sum Function

The sum function S(r) = 1/(1 - r) is defined for all r except r = 1, where it diverges. Its range depends on the values of r, especially considering convergence criteria. Analyzing the interval of r values that produce sums within (1/2, ∞) helps determine if the series covers that entire interval.

Convergence Criteria for Infinite Series

For an infinite geometric series to converge, the common ratio r must satisfy |r| < 1. If |r| ≥ 1, the series diverges and does not sum to a finite value. This criterion restricts the possible sums and is crucial when evaluating whether the series can take all values in a given interval.

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.g. Viewed as a function of r, the series 1 + r + r² + r³ + ⋯ takes on all values in the interval (1/2, ∞).