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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.71a
Capitolo 10, Problema 10.3.71a

71. Evaluating an infinite series two ways
Evaluate the series
∑ (k = 1 to ∞) (4 / 3ᵏ – 4 / 3ᵏ⁺¹) two ways.
a. Use a telescoping series argument.

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Write out the general term of the series explicitly: \(a_k = \frac{4}{3^k} - \frac{4}{3^{k+1}}\).
Recognize that this is a telescoping series because each term can be rewritten to show cancellation between consecutive terms when summed.
Express the partial sum \(S_n = \sum_{k=1}^n \left( \frac{4}{3^k} - \frac{4}{3^{k+1}} \right)\) and write out the first few terms to observe the telescoping pattern.
Notice that most terms cancel out, leaving only the first term of the first fraction and the last term of the second fraction in the partial sum.
Write the simplified form of \(S_n\) after cancellation and then take the limit as \(n \to \infty\) to find the sum of the infinite series.

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Telescoping Series

A telescoping series is a series where many terms cancel out when the partial sums are expanded, leaving only a few terms to evaluate. This simplification makes it easier to find the sum of the infinite series by focusing on the first and last terms of the partial sums.
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Geometric Series

Infinite Geometric Series

An infinite geometric series has terms that multiply by a constant ratio each time. If the absolute value of the ratio is less than one, the series converges, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.
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Geometric Series

Partial Sums and Convergence

Partial sums are the sums of the first n terms of a series. Understanding how these sums behave as n approaches infinity helps determine if the series converges and what its sum is. For telescoping and geometric series, analyzing partial sums is key to evaluating the infinite sum.
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Integration Using Partial Fractions
Pratica correlata
Domanda del libro di testo

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. If the Limit Comparison Test can be applied successfully to a given series with a certain comparison series, the Comparison Test also works with the same comparison series.

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

a. Suppose 0 < aₖ < bₖ. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 1 to ∞) bₖ converges.

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57–60. Heights of bouncing balls A ball is thrown upward to a height of hₒ meters. After each bounce, the ball rebounds to a fraction r of its previous height. Let hₙ be the height after the nth bounce. Consider the following values of hₒ and r.


a. Find the first four terms of the sequence of heights {hₙ}.


h₀ = 20,r = 0.5

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Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.


a.The sequence of partial sums for the series1 + 2 + 3 + ⋯ is {1, 3, 6, 10, …}.

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


a. A series that converges must converge absolutely.

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72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence


a.Write out the first five terms of the sequence.


Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

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