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


b. If ∑ (k = 1 to ∞) aₖ diverges, then ∑ (k = 10 to ∞) aₖ diverges.

Verified step by step guidance

1. Understand the problem: We are asked to determine if the statement "If \( \sum_{k=1}^{\infty} a_k \) diverges, then \( \sum_{k=10}^{\infty} a_k \) also diverges" is true or false.
2. Recall the definition of series convergence and divergence: A series \( \sum_{k=m}^{\infty} a_k \) converges if the sequence of partial sums \( S_n = \sum_{k=m}^n a_k \) approaches a finite limit as \( n \to \infty \). Otherwise, it diverges.
3. Analyze the relationship between the two series: The series starting at \( k=10 \) is essentially the tail of the series starting at \( k=1 \). The original series can be written as \( \sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k \).
4. Consider the impact of the finite sum \( \sum_{k=1}^{9} a_k \): Since this is a finite sum, it does not affect convergence or divergence of the infinite series. Therefore, the convergence or divergence of \( \sum_{k=10}^{\infty} a_k \) determines the behavior of the tail.
5. Conclude based on the above: If the entire series \( \sum_{k=1}^{\infty} a_k \) diverges, then its tail \( \sum_{k=10}^{\infty} a_k \) must also diverge, because adding or removing a finite number of terms does not change the divergence property.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms from a sequence. A series converges if its partial sums approach a finite limit; otherwise, it diverges. Understanding convergence is essential to analyze whether changing the starting index affects the series' behavior.

Effect of Changing the Index of Summation

Shifting the starting index of a series changes only a finite number of terms. Since convergence depends on the tail behavior of the series, adding or removing finitely many terms does not affect whether the series converges or diverges.

Counterexamples in Series Analysis

Counterexamples demonstrate when a general statement is false. To test if a series starting at k=10 diverges when the series from k=1 diverges, one can consider series where initial terms affect convergence, highlighting the importance of examining specific cases.