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 10.3.65
Capitolo 10, Problema 10.3.65

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.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

Guida verificata passo dopo passo
1
Identify the general term of the series: \( a_k = \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+3}} \). This is a telescoping series because each term involves the difference of two expressions with consecutive indices shifted by 2.
Write the nth partial sum \( S_n \) as the sum of the first n terms: \[ S_n = \sum_{k=1}^n \left( \frac{1}{\sqrt{k+1}} - \frac{1}{\sqrt{k+3}} \right) \].
Expand the sum to observe cancellation: \[ S_n = \left( \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{4}} \right) + \left( \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{5}} \right) + \cdots + \left( \frac{1}{\sqrt{n+1}} - \frac{1}{\sqrt{n+3}} \right) \].
Group the positive and negative terms to see which terms cancel out. Notice that many intermediate terms will cancel, leaving only a few terms from the beginning and the end of the sum.
Express \( S_n \) in a simplified form involving only the first few terms and the last few terms that do not cancel. Then, evaluate \( \lim_{n \to \infty} S_n \) by analyzing the behavior of the remaining terms as \( n \) approaches infinity to determine if the series converges or diverges.

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:
4m

Concetti chiave

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

Telescoping Series

A telescoping series is a series whose partial sums simplify because many terms cancel out. Typically, each term can be written as the difference of two expressions, causing intermediate terms to vanish when summed. This simplification helps find a closed-form expression for the nth partial sum.
Video consigliato:
Percorso guidato
06:00
Geometric Series

Partial Sums and Their Limits

The partial sum Sₙ of a series is the sum of its first n terms. Evaluating the limit of Sₙ as n approaches infinity determines whether the series converges or diverges. If the limit exists and is finite, the series converges to that value.
Video consigliato:
Percorso guidato
08:01
Integration Using Partial Fractions

Manipulating Radical Expressions

Working with terms involving square roots often requires rationalizing or rewriting expressions to simplify differences. Recognizing patterns in radicals can help express terms in a form suitable for telescoping, making it easier to identify cancellations in the series.
Video consigliato:
06:13
Limits of Rational Functions with Radicals
Pratica correlata
Domanda del libro di testo

The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.

n an

1 2.59374246

2 2.70481383

3 2.71692393

4 2.71814593

5 2.71826824

6 2.71828047

7 2.71828169

8 2.71828179

9 2.71828204

10 2.71828203

60
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 ∞) tan(1 / k)

59
views
Domanda del libro di testo

45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

47
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 k! / kᵏ

46
views
Domanda del libro di testo

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

∑ (from j = 2 to ∞)1 / (j ln¹⁰j)

51
views
Domanda del libro di testo

"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 3aₙ-12; a₁ = 10

76
views