Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.

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Recall the p-series test: The series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if and only if \( p > 1 \).

Given that \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges, it implies that \( p > 1 \).

Now consider the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \). Since \( p + 0.001 > p > 1 \), this new exponent is also greater than 1.

By the p-series test again, since the exponent \( p + 0.001 \) is greater than 1, the series \( \sum_{k=1}^{\infty} \frac{1}{k^{p + 0.001}} \) also converges.

Therefore, the statement is true because increasing the exponent in a convergent p-series to a slightly larger value still results in a convergent series.

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주요 개념

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

p-Series Test for Convergence

The p-series test states that the series ∑ 1/k^p converges if and only if p > 1. This test helps determine whether a series with terms involving powers of k converges or diverges based on the value of p.

Comparison Test for Series

The comparison test allows us to determine convergence by comparing a given series to another series with known behavior. If a series with larger terms converges, then a series with smaller terms also converges, and vice versa.

Effect of Small Changes in the Exponent on Convergence

Slightly increasing the exponent p in a p-series (e.g., from p to p + 0.001) affects convergence because the series terms decrease faster. If the original series converges, increasing p will also result in convergence, but if it diverges, the new series may still diverge.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.e. If ∑ k⁻ᵖ converges, then ∑ k⁻ᵖ⁺⁰.⁰⁰¹ converges.