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.

Verified step by step guidance

1. Recall the p-series test: The series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if and only if \( p > 1 \).
2. Given that \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges, it implies that \( p > 1 \).
3. 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.
4. 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.
5. 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.

Key Concepts

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.