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

f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.

검증된 단계별 안내

Step 1: Understand the problem statement. We are given that the series \( \sum_{k=1}^\infty a^k \) converges and that \( |a| < |b| \). We need to determine if this implies that the series \( \sum_{k=1}^\infty b^k \) also converges.

Step 2: Recall the convergence criteria for geometric series. A geometric series \( \sum_{k=1}^\infty r^k \) converges if and only if \( |r| < 1 \). If \( |r| \geq 1 \), the series diverges.

Step 3: Since \( \sum_{k=1}^\infty a^k \) converges, it follows that \( |a| < 1 \). This is because the series is geometric with ratio \( a \).

Step 4: Given \( |a| < |b| \), and knowing \( |a| < 1 \), it is possible that \( |b| \geq 1 \). If \( |b| \geq 1 \), then the series \( \sum_{k=1}^\infty b^k \) diverges by the geometric series test.

Step 5: Therefore, the statement is not necessarily true. A counterexample would be choosing \( a = 0.5 \) and \( b = 0.8 \) (both less than 1, so both series converge), but if \( b = 1.2 \), then \( |a| < |b| \) but \( \sum b^k \) diverges. Hence, the statement is false.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. For series involving powers, such as ∑ a^k, convergence depends on the value of the base a. Understanding convergence criteria is essential to determine whether a given series sums to a finite value.

Geometric Series and Their Convergence

A geometric series ∑ r^k converges if and only if the absolute value of the common ratio |r| is less than 1. If |r| ≥ 1, the series diverges. This concept is crucial for analyzing series of the form ∑ a^k and ∑ b^k, as their convergence depends directly on the magnitude of a and b.

Comparison of Series Based on Term Magnitudes

Comparing series by the size of their terms can help infer convergence properties. However, a larger base in a geometric series does not guarantee convergence if the smaller base series converges. Understanding that convergence depends on the absolute value of the base, not just relative size, is key to evaluating the given statement.

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.f. If the series ∑ (k = 1 to ∞) aᵏ converges and |a| < |b|, then the series ∑ (k = 1 to ∞) bᵏ converges.