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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
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.3.33
Capitolo 10, Problema 10.3.33

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


33.∑ (k = 4 to ∞) 1 / 5ᵏ

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Identify the first term \( a \) of the geometric series by substituting the starting index \( k = 4 \) into the term \( \frac{1}{5^k} \). So, \( a = \frac{1}{5^4} \).
Determine the common ratio \( r \) of the geometric series. Since the terms are of the form \( \frac{1}{5^k} \), the ratio between consecutive terms is \( \frac{1}{5} \).
Check the convergence of the series by verifying if the absolute value of the common ratio \( |r| < 1 \). If this condition holds, the series converges; otherwise, it diverges.
If the series converges, use the formula for the sum of an infinite geometric series starting at \( k = 0 \): \[ S = \frac{a}{1 - r} \]. Since our series starts at \( k = 4 \), \( a \) is already the first term at \( k=4 \).
Substitute the values of \( a \) and \( r \) into the sum formula to express the sum of the series. This will give the sum in terms of powers of 5 without calculating the final numeric value.

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Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, where a is the first term and r is the common ratio. Understanding this structure helps in identifying and evaluating the series.
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Geometric Series

Convergence of Infinite Geometric Series

An infinite geometric series converges if the absolute value of the common ratio |r| is less than 1. When it converges, the sum can be calculated using the formula S = a / (1 - r). If |r| ≥ 1, the series diverges and does not have a finite sum.
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Convergence of an Infinite Series

Index Shift in Series Summation

When a series starts at an index other than zero or one, it may be necessary to adjust the formula for the sum by rewriting the series with a shifted index. This helps in correctly identifying the first term a and applying the geometric series sum formula.
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Geometric Series
Pratica correlata
Domanda del libro di testo

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

∑ (from k = 1 to ∞)5¹⁻²ᵏ

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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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Domanda del libro di testo

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.


∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))

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Domanda del libro di testo

13–52. Limits of sequences

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


{bₙ}, where

bₙ = { n / (n + 1)if n ≤ 5000

ne⁻ⁿif n > 5000 }

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Domanda del libro di testo

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

∑ (k = 1 to ∞) (−1)ᵏ⁺¹ k¹/ᵏ

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Domanda del libro di testo

21–42. Geometric series Evaluate each geometric series or state that it diverges.  


29.∑ (k = 1 to ∞) e^(–2k)

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