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

d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.

검증된 단계별 안내

Recall the behavior of geometric series: For a fixed real number \( r \), the series \( \sum r^k \) converges if and only if \( |r| < 1 \), and diverges otherwise.

Given that \( \sum p^k \) diverges, this means that \( |p| \geq 1 \). This is the key starting point for analyzing the series \( \sum (p + 0.001)^k \).

Consider the value of \( |p + 0.001| \). Since \( p \) is fixed, adding 0.001 shifts the base slightly. We need to determine if this new base still satisfies \( |p + 0.001| \geq 1 \) or not.

If \( |p + 0.001| \geq 1 \), then \( \sum (p + 0.001)^k \) also diverges by the geometric series test. However, if \( |p + 0.001| < 1 \), then \( \sum (p + 0.001)^k \) converges, providing a counterexample to the statement.

Therefore, to conclude whether the statement is true or false, analyze specific values of \( p \) where \( |p| = 1 \) and see how the small increment affects convergence or divergence.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Geometric Series and Convergence Criteria

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 criterion is fundamental to analyzing series of the form ∑ p^k and ∑ (p + 0.001)^k.

Effect of Changing the Common Ratio on Series Convergence

Altering the common ratio by a small amount (e.g., adding 0.001) can change whether the series converges or diverges. Even a slight increase can push the ratio beyond the convergence boundary, affecting the series' behavior significantly.

Counterexamples in Series Convergence

To determine the truth of a statement about series convergence, constructing counterexamples is essential. For instance, if ∑ p^k diverges but ∑ (p + 0.001)^k converges, this disproves the statement. Counterexamples help clarify the limits of general claims.

87. Explain why or why notDetermine whether the following statements are true and give an explanation or counterexample.d. If ∑ pᵏ diverges, then ∑ (p + 0.001)ᵏ diverges, for a fixed real number p.