Skip to main content
Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.3.64
Capitolo 10, Problema 10.3.64

Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.

Guida verificata passo dopo passo
1
Identify the function corresponding to the terms of the series: define \( f(x) = e^{-x^2} \). This function is positive and continuous for all \( x \geq 0 \).
Check if \( f(x) \) is decreasing for \( x \geq 0 \). To do this, compute the derivative \( f'(x) \) and verify that it is negative for \( x > 0 \).
Set up the improper integral \( \int_0^{\infty} e^{-x^2} \, dx \) to apply the Integral Test. This integral is known as the Gaussian integral (or related to it).
Determine whether the integral \( \int_0^{\infty} e^{-x^2} \, dx \) converges. Since this integral converges to a finite value, it implies the series converges by the Integral Test.
Conclude that because \( f(x) = e^{-x^2} \) is positive, continuous, decreasing for \( x \geq 0 \), and the integral of \( f(x) \) from 0 to infinity converges, the series \( \sum_{n=0}^{\infty} e^{-n^2} \) converges.

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:
5m

Concetti chiave

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

Integral Test for Series Convergence

The Integral Test determines the convergence of an infinite series by comparing it to an improper integral. If the function corresponding to the series terms is positive, continuous, and decreasing, then the series converges if and only if the integral of the function from some point to infinity converges.
Video consigliato:
Percorso guidato
07:51
Choosing a Convergence Test

Behavior of the Function e^(−n²)

The function e^(−n²) decreases rapidly as n increases because the exponent is negative and grows quadratically. This rapid decay suggests the terms approach zero quickly, which is a key factor in the convergence of the series.
Video consigliato:
4:47
The Number e

Improper Integrals and Their Convergence

An improper integral extends over an infinite interval or involves an unbounded integrand. To apply the Integral Test, one evaluates the integral of e^(−x²) from a finite point to infinity, which is known to converge due to the Gaussian integral properties.
Video consigliato:
Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Pratica correlata
Domanda del libro di testo

Determining Convergence or Divergence

In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑(from n=1 to ∞) [(2 + (−1)ⁿ) / 1.25ⁿ]

Domanda del libro di testo

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [(2n + 3)(2ⁿ + 3) / (3ⁿ + 2)]

Domanda del libro di testo

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 2 to ∞) [(3n)! / (n!)³]

Domanda del libro di testo

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

56
views
Domanda del libro di testo

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (n + 3) / (n² + 5n + 6)

29
views
Domanda del libro di testo

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.

25
views