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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.2.83c
Chapter 10, Problem 10.2.83c

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


c.The convergent sequences {aₙ} and {bₙ} differ in their first 100 terms, but aₙ = bₙ for n > 100.
It follows that limₙ→∞aₙ = limₙ→∞bₙ.

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1
Recall the definition of the limit of a sequence: A sequence \( \{a_n\} \) converges to a limit \( L \) if for every \( \varepsilon > 0 \), there exists an \( N \) such that for all \( n > N \), \( |a_n - L| < \varepsilon \).
Note that the behavior of a sequence at finitely many terms (such as the first 100 terms) does not affect the limit, because the limit depends only on the terms as \( n \to \infty \).
Given that \( a_n = b_n \) for all \( n > 100 \), the two sequences are identical from the 101st term onward.
Since both sequences \( \{a_n\} \) and \( \{b_n\} \) are convergent, and they agree on all terms beyond \( n = 100 \), their tails are the same and must converge to the same limit.
Therefore, it follows that \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} b_n \), regardless of differences in the first 100 terms.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of Convergent Sequences

A sequence {aₙ} converges to a limit L if, for every positive ε, there exists an N such that for all n > N, |aₙ - L| < ε. This means the terms get arbitrarily close to L as n becomes large, regardless of the initial terms.
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Effect of Finite Differences on Limits

Changing a finite number of terms in a sequence does not affect its limit. Since limits depend on the behavior of terms as n approaches infinity, differences in the first 100 terms do not alter the limit of the sequence.
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Uniqueness of Limits for Sequences

If two sequences are equal from some index onward and both converge, they must share the same limit. This follows because their tails coincide, so their limiting behavior is identical.
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Related Practice
Textbook Question

{Use of Tech} Periodic dosing

Many people take aspirin on a regular basis as a preventive measure for heart disease. Suppose a person takes 80 mg of aspirin every 24 hours. Assume aspirin has a half-life of 24 hours; that is, every 24 hours, half of the drug in the blood is eliminated.


c.Assuming the sequence has a limit, confirm the result of part (b) by finding the limit of {dₙ} directly.

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Textbook Question

39–40. {Use of Tech} Lower and upper bounds of a series

For each convergent series and given value of n, use Theorem 10.13 to complete the following.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) for the exact value of the series.


39. ∑ (k = 1 to ∞) 1 / k⁷ ; n = 2

Textbook Question

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


c. Suppose f is a continuous, positive, decreasing function, for x ≥ 1, and aₖ = f(k), for k = 1, 2, 3, …. If ∑ (k = 1 to ∞) aₖ converges to L, then ∫ (1 to ∞) f(x) dx converges to L.

Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.


{1, 3, 9, 27, 81, ......}

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

c. Find an explicit formula for the nth term of the sequence.


{-5, 5, -5, 5, ......}

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Textbook Question

41–44. {Use of Tech} Remainders and estimates Consider the following convergent series.


c. Find lower and upper bounds (Lₙ and Uₙ, respectively) on the exact value of the series.


41. ∑ (k = 1 to ∞) 1 / k⁶