Domanda del libro di testo
13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)
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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 10.2.31
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(3ⁿ⁺¹ + 3)⁄3ⁿ}
Guida verificata passo dopo passo1
Identify the given sequence: \(a_n = \frac{3^{n+1} + 3}{3^n}\).
Rewrite the expression to simplify it by separating the terms in the numerator: \(a_n = \frac{3^{n+1}}{3^n} + \frac{3}{3^n}\).
Simplify each term using the properties of exponents: \(\frac{3^{n+1}}{3^n} = 3^{(n+1)-n} = 3^1 = 3\) and \(\frac{3}{3^n} = 3 \cdot 3^{-n} = 3^{1-n}\).
Express the sequence as \(a_n = 3 + 3^{1-n}\) and analyze the behavior of \(3^{1-n}\) as \(n\) approaches infinity.
Since \(3^{1-n} = \frac{3}{3^n}\) and \$3^n$ grows without bound as \(n \to \infty\), \(3^{1-n} \to 0\). Therefore, the limit of the sequence is \(\lim_{n \to \infty} a_n = 3 + 0 = 3\).
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Limits of Sequences
The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges; otherwise, it diverges. Understanding this helps determine the behavior of sequences like (3ⁿ⁺¹ + 3)/3ⁿ as n grows large.
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8:22
Introduction to Sequences
Properties of Exponents
Exponents follow specific rules such as a^(m+n) = a^m * a^n and a^m / a^n = a^(m-n). Applying these properties simplifies expressions involving powers, which is essential for rewriting and analyzing sequences like (3ⁿ⁺¹ + 3)/3ⁿ to find their limits.
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Percorso guidato
06:21Properties of Functions
Algebraic Simplification
Algebraic simplification involves rewriting expressions in simpler or more convenient forms. For sequences, this often means factoring or dividing terms to isolate dominant parts, making it easier to evaluate limits. Simplifying (3ⁿ⁺¹ + 3)/3ⁿ helps identify the dominant term as n approaches infinity.
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05:25Determine Continuity Algebraically
Pratica correlata
Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) (2k⁴ + k) / (4k⁴ − 8k)
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Domanda del libro di testo
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
aₙ = (−1)ⁿ ⁿ√n
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Domanda del libro di testo
11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞)(⁵√k) / ⁵√(k⁷ + 1)
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Domanda del libro di testo
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹n / n}
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Domanda del libro di testo
13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(1 + (2 / n))ⁿ}
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