Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.5.63

Chapter 10, Problem 10.5.63

Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

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Recall that the series \( \sum a_k \) converges and each term \( a_k > 0 \). Since the terms are positive and the series converges, it follows that \( a_k \to 0 \) as \( k \to \infty \).

Because \( a_k \to 0 \), there exists an index \( N \) such that for all \( k > N \), \( a_k < 1 \). This will help us compare \( a_k^2 \) to \( a_k \) for large \( k \).

For \( k > N \), since \( 0 < a_k < 1 \), we have \( a_k^2 < a_k \). This inequality allows us to use the Comparison Test for series convergence.

Apply the Comparison Test: since \( \sum a_k \) converges and \( a_k^2 < a_k \) for all \( k > N \), the tail \( \sum_{k=N+1}^\infty a_k^2 \) converges by comparison.

Finally, since the finite sum \( \sum_{k=1}^N a_k^2 \) is finite and the tail \( \sum_{k=N+1}^\infty a_k^2 \) converges, the entire series \( \sum a_k^2 \) converges.

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Key Concepts

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

Convergent Series of Positive Terms

A series ∑aₖ with positive terms converges if the sequence of partial sums approaches a finite limit. Since all terms are positive, the partial sums form a non-decreasing sequence bounded above, ensuring convergence. Understanding this helps establish the behavior of the original series.

Comparison Test for Series

The comparison test states that if 0 ≤ aₖ ≤ bₖ for all k and ∑bₖ converges, then ∑aₖ also converges. This test is useful to compare the series of squares ∑aₖ² with the original series ∑aₖ or another known convergent series to prove convergence.

Behavior of Squares of Terms in a Convergent Series

If ∑aₖ converges with positive terms, then aₖ → 0 as k → ∞. For sufficiently large k, aₖ < 1, so aₖ² < aₖ. This inequality allows us to use the comparison test to show that ∑aₖ² converges by comparing it to ∑aₖ.

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Related Practice

Textbook Question

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{sinn / 2ⁿ}

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Textbook Question

84–87. {Use of Tech} Sequences by recurrence relations

The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.

a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.

b.Use analytical methods to find the limit of the sequence.

{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3

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Textbook Question

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) (5 / 6)⁻ᵏ

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 10 to ∞) 1 / (k − 9)⁵

Textbook Question

Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k.

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