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Chapter 9: Problem 8

Explain how two sequences that differ only in their first ten terms can have the same limit.

Short Answer

Expert verified
Explain how two sequences that differ only in their first ten terms can have the same limit. Two sequences \((a_n)\) and \((b_n)\) can have the same limit, even if they differ in their first ten terms, because the limit is only concerned with their behavior as the number of terms increases. The first ten terms do not affect the convergence of these sequences in the long run. After the 10th term, the two sequences share the same terms, and thus, their convergence properties are identical. Therefore, by definition, they will have the same limit, \(L\).

Step by step solution

01

1. Understand the definition of a limit.

A sequence has a limit if it approaches a certain value as the number of terms increases. More formally, a sequence \((a_n)\) has a limit \(L\) if for every positive number \(\epsilon\) there exists a natural number \(N\) such that for all \(n>N\), we have \(|a_n - L| < \epsilon\).
02

2. Consider two sequences that differ only in their first ten terms.

Let \((a_n)\) and \((b_n)\) be two sequences such that \(a_n = b_n\) for \(n > 10\), but \(a_n \neq b_n\) for \(n \leq 10\). This means that the sequences are different for the first ten terms but become identical from the 11th term onwards.
03

3. Analyze the properties of the limit for the two sequences.

Assume that both sequences converge to the same limit \(L\). According to the definition of a limit, for any \(\epsilon > 0\), there exists a natural number \(N_a\) (for \((a_n)\)) and \(N_b\) (for \((b_n)\)) such that for all \(n>N_a\), \(|a_n - L| < \epsilon\) and for all \(n>N_b\), \(|b_n - L| < \epsilon\).
04

4. Illustrate the condition for the sequences to have the same limit.

Since both \((a_n)\) and \((b_n)\) share common terms from the 11th term onwards, we can choose a value of \(N\) which is greater than 10, \(N_a\), and \(N_b\). Let's call this value \(N^*\). We know that for every \(n > N^*\), the differences between both \(a_n\) and \(b_n\) from the limit \(L\) are strictly less than \(\epsilon\).
05

5. Concluding the explanation.

The condition for two sequences to have the same limit is satisfied in this case, because the first ten terms do not affect the convergence of the sequences in the long run. Since the terms of both sequences are identical after the 10th term, they share the same properties concerning convergence and, by definition, will have the same limit, \(L\).

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