Domanda del libro di testo
42–76. Convergence or divergence Use a convergence test of your choice to determine whether the following series converge.
∑ (from k = 1 to ∞)k⁴ / √(9k¹² + 2)
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Ch. 10 - Sequences and Infinite Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 10, Problema 10.R.1e
Explain why or why not
Determine whether the following statements are true and give an explanation or counterexample.
e.The sequence aₙ = n² / (n² + 1) converge.
Guida verificata passo dopo passo1
Identify the sequence given: \(a_n = \frac{n^2}{n^2 + 1}\).
Recall the definition of convergence for a sequence: a sequence \(a_n\) converges to a limit \(L\) if \(\lim_{n \to \infty} a_n = L\) exists and is finite.
To determine if \(a_n\) converges, compute the limit as \(n\) approaches infinity: \(\lim_{n \to \infty} \frac{n^2}{n^2 + 1}\).
Divide numerator and denominator by \(n^2\) to simplify the expression: \(\lim_{n \to \infty} \frac{1}{1 + \frac{1}{n^2}}\).
Evaluate the limit by noting that \(\frac{1}{n^2} \to 0\) as \(n \to \infty\), so the limit becomes \(\frac{1}{1 + 0} = 1\), which means the sequence converges to 1.
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Sequence Convergence
A sequence converges if its terms approach a specific finite value as the index n approaches infinity. This means for any small positive number, the terms eventually get arbitrarily close to that value. Understanding convergence helps determine the long-term behavior of sequences.
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Introduction to Sequences
Limit of a Sequence
The limit of a sequence is the value that the terms approach as n becomes very large. Calculating the limit often involves simplifying the expression and analyzing dominant terms. If the limit exists and is finite, the sequence converges to that limit.
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8:22Introduction to Sequences
Dominant Term Analysis
When evaluating limits of sequences involving polynomials, the highest degree terms dominate the behavior as n grows large. By comparing the degrees of numerator and denominator, one can simplify the expression to find the limit, which is crucial for determining convergence.
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Percorso guidato
05:44Divergence Test (nth Term Test)
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