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 10.6.3

Capitolo 10, Problema 10.6.3

Why does the value of a converging alternating series with terms that are nonincreasing in magnitude lie between any two consecutive terms of its sequence of partial sums?

Guida verificata passo dopo passo

Recall the Alternating Series Test, which states that if the terms of an alternating series decrease in magnitude to zero, then the series converges.

Understand that the sequence of partial sums of such a series alternates above and below the actual sum because each new term added changes the partial sum by a smaller and smaller amount, switching signs each time.

Note that since the terms are nonincreasing in magnitude, the difference between consecutive partial sums decreases, causing the partial sums to 'trap' the true sum between them.

Recognize that this trapping means the true sum lies between any two consecutive partial sums, because the partial sums form an alternating sequence converging to the sum from above and below.

Conclude that this property ensures the value of the converging alternating series is always bounded between any two consecutive partial sums, providing a natural error bound for approximations.

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Alternating Series

An alternating series is a series whose terms alternate in sign, typically written as positive and negative terms in succession. This pattern affects the behavior of partial sums, causing them to oscillate above and below the series' limit, which is key to understanding convergence properties.

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06:00

Geometric Series

Monotone Decreasing Terms in Magnitude

When the absolute values of the terms in a series decrease monotonically (each term is smaller or equal in magnitude than the previous), it ensures the partial sums approach the limit in a controlled manner. This condition is crucial for applying the Alternating Series Test and for bounding the error between partial sums and the series sum.

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05:44

Divergence Test (nth Term Test)

Partial Sums and Convergence Bounds

Partial sums are the sums of the first n terms of a series. For converging alternating series with decreasing term magnitudes, the sequence of partial sums alternates around the limit, meaning the true sum lies between any two consecutive partial sums. This property provides a natural bound on the approximation error.

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Integration Using Partial Fractions

Pratica correlata

Domanda del libro di testo

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.

aₙ = 1/10ⁿ

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23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.

∑ (k = 1 to ∞) k^(1/k)

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from j = 1 to ∞) 5 / (j² + 4)

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54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.

61. ∑ (k = 1 to ∞) ln((k + 1) / k)

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39–44. {Use of Tech} Estimating infinite series Estimate the value of the following convergent series with an absolute error less than 10⁻³.

∑ (k = 1 to ∞) (−1)ᵏ / k⁵

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Domanda del libro di testo

17–22. Integral Test Use the Integral Test to determine whether the following series converge after showing that the conditions of the Integral Test are satisfied.

∑ (k = 1 to ∞) 1 / (∛(5k + 3))

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