Skip to main content
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 10.3.47
Capitolo 10, Problema 10.3.47

46–53. Decimal expansions
Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers).
47.0.3̅ = 0.333…

Guida verificata passo dopo passo
1
Identify the repeating decimal: here it is \(0.3\overline{3} = 0.333\ldots\) where the digit 3 repeats indefinitely.
Express the repeating decimal as an infinite geometric series. The first term is \(\frac{3}{10}\) (since 3 is in the tenths place), and each subsequent term is multiplied by \(\frac{1}{10}\) because the decimal repeats every place value. So the series is: \(\frac{3}{10} + \frac{3}{10^2} + \frac{3}{10^3} + \cdots\).
Recognize this as a geometric series with first term \(a = \frac{3}{10}\) and common ratio \(r = \frac{1}{10}\).
Use the formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(|r| < 1\).
Substitute \(a = \frac{3}{10}\) and \(r = \frac{1}{10}\) into the formula to write the decimal as a fraction: \(S = \frac{\frac{3}{10}}{1 - \frac{1}{10}}\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Repeating Decimals

A repeating decimal is a decimal number in which a digit or group of digits repeats infinitely. For example, 0.3̅ means 0.333..., where the digit 3 repeats endlessly. Understanding this helps in converting such decimals into exact fractions.
Video consigliato:
Percorso guidato
08:23
Repeated Integration by Parts

Geometric Series

A geometric series is a sum of terms where each term is found by multiplying the previous term by a constant ratio. Repeating decimals can be expressed as infinite geometric series, where each term represents a decimal place value.
Video consigliato:
Percorso guidato
06:00
Geometric Series

Conversion of Infinite Geometric Series to Fractions

An infinite geometric series with first term 'a' and common ratio 'r' (|r|<1) converges to a/(1-r). This formula allows us to convert the geometric series representation of a repeating decimal into a fraction, providing an exact ratio of two integers.
Video consigliato:
Percorso guidato
06:00
Geometric Series
Pratica correlata
Domanda del libro di testo

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

79
views
Domanda del libro di testo

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) k⁸ / (k¹¹ + 3)

40
views
Domanda del libro di testo

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

1 / (2·3) + 1 / (4·5) + 1 / (6·7) + 1 / (8·9) + ⋯

27
views
Domanda del libro di testo

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

aₙ = (−1)ⁿ / 2ⁿ

47
views
Domanda del libro di testo

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


37.1 + e/π + e²/π² + e³/π³ + ⋯

59
views
Domanda del libro di testo

32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) (2k + 1)! / (k!)²

35
views