Domanda del libro di testo
Use the Integral Test to show that the series
∑ (from n=0 to ∞) e^(−n²)
converges.
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 10, Problema 10.5.10
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]
Guida verificata passo dopo passo1
Identify the general term of the series: \(a_n = \frac{4^n}{(3n)^n}\).
Recall the Root Test formula: compute \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).
Apply the nth root to the term: \(\sqrt[n]{|a_n|} = \sqrt[n]{\frac{4^n}{(3n)^n}} = \frac{\sqrt[n]{4^n}}{\sqrt[n]{(3n)^n}}\).
Simplify the nth roots: \(\frac{4}{3n}\).
Evaluate the limit as \(n\) approaches infinity: \(L = \lim_{n \to \infty} \frac{4}{3n}\). Use this value to determine convergence: if \(L < 1\), the series converges absolutely; if \(L > 1\), it diverges; if \(L = 1\), the test is inconclusive.
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Root Test for Series Convergence
The Root Test determines the convergence of an infinite series by examining the nth root of the absolute value of its terms. Specifically, if the limit of the nth root of |a_n| as n approaches infinity is less than 1, the series converges absolutely; if greater than 1, it diverges; and if equal to 1, the test is inconclusive.
Video consigliato:
Percorso guidato
07:15
Root Test
Absolute Convergence
A series converges absolutely if the series of the absolute values of its terms converges. Absolute convergence guarantees convergence regardless of the sign of terms, making it a stronger form of convergence and simplifying the analysis of series with alternating or complex terms.
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Percorso guidato
07:51Choosing a Convergence Test
Evaluating Limits Involving nth Roots
To apply the Root Test, one must compute the limit of the nth root of the series terms. This often involves simplifying expressions with powers and roots, such as rewriting terms like (4^n / (3n)^n)^(1/n) into a more manageable form to find the limit as n approaches infinity.
Video consigliato:
06:13Limits of Rational Functions with Radicals
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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 ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]
(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)
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Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
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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)
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Find the value of b for which
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