Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.5.63

Capitolo 10, Problema 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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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.

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Percorso guidato

06:52

Convergence of an Infinite 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.

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Percorso guidato

09:25

Direct Comparison Test

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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Percorso guidato

06:52

Convergence of an Infinite Series

Pratica correlata

Domanda del libro di testo

55–70. More sequences

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

{sinn / 2ⁿ}

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

∑ (k = 2 to ∞) (−1)ᵏ (1 + 1/k)

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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

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46–53. Decimal expansions

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).

53.0.00952̅ = 0.00952952…

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Domanda del libro di testo

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.

2 / 4² + 2 / 5² + 2 / 6² + ⋯

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