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


a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

Verified step by step guidance

1. Understand the problem: We are asked to determine if the convergence of the infinite series \(\sum_{k=1}^{\infty} a_k\) implies the convergence of the series \(\sum_{k=10}^{\infty} a_k\).
2. Recall the definition of convergence for infinite series: 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 \(n \to \infty\).
3. Consider the relationship between the two series: The series starting at \(k=10\) is essentially the tail of the series starting at \(k=1\). Specifically, \(\sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k\).
4. Since the sum of the first 9 terms, \(\sum_{k=1}^{9} a_k\), is a finite number, subtracting it from the convergent series \(\sum_{k=1}^{\infty} a_k\) leaves the tail \(\sum_{k=10}^{\infty} a_k\), which must also converge.
5. Therefore, the convergence of \(\sum_{k=1}^{\infty} a_k\) guarantees the convergence of \(\sum_{k=10}^{\infty} a_k\) because removing a finite number of terms from the start of a convergent series does not affect its convergence.

Key Concepts

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. A series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether changing the starting index affects the sum's behavior.

Effect of Finite Number of Terms on Convergence

Adding or removing a finite number of terms from an infinite series does not affect its convergence. Since convergence depends on the tail behavior of the series, starting the sum at k=10 instead of k=1 preserves convergence if the original series converges.

Partial Sums and Tail of a Series

Partial sums are sums of the first n terms of a series. The tail of a series refers to the sum from some index onward. Convergence depends on the tail's limit, so analyzing the tail from k=10 onward helps determine if the series remains convergent.