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.

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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 \(\sum_{k=10}^{\infty} a_k\).

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 \(n \to \infty\).

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, \(\sum_{k=1}^{\infty} a_k = \sum_{k=1}^{9} a_k + \sum_{k=10}^{\infty} a_k\).

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.

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.

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