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

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.2.11

Capitolo 10, Problema 10.2.11

Compare the growth rates of {n¹⁰⁰} and {eⁿ⁄¹⁰⁰} as n → ∞.

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Identify the two functions to compare: \(f(n) = n^{100}\) and \(g(n) = e^{\frac{n}{100}}\).

Recall that to compare growth rates as \(n \to \infty\), we often consider the limit of their ratio, such as \(\lim_{n \to \infty} \frac{f(n)}{g(n)}\) or \(\lim_{n \to \infty} \frac{g(n)}{f(n)}\).

Set up the limit \(L = \lim_{n \to \infty} \frac{n^{100}}{e^{\frac{n}{100}}}\) to analyze which function grows faster.

Apply L'Hôpital's Rule if the limit is an indeterminate form like \(\frac{\infty}{\infty}\) by differentiating numerator and denominator with respect to \(n\) repeatedly, or use properties of exponential and polynomial functions to reason about the limit.

Conclude the comparison based on the limit: if \(L = 0\), then \(g(n)\) grows faster; if \(L = \infty\), then \(f(n)\) grows faster; if \(L\) is a finite nonzero constant, they grow at comparable rates.

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Exponential Growth

Exponential growth refers to functions where the variable appears in the exponent, such as e^(n/100). These functions increase much faster than polynomial functions as n becomes very large, often dominating growth rates in limits approaching infinity.

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Exponential Growth & Decay

Polynomial Growth

Polynomial growth involves functions where the variable is raised to a fixed power, like n^100. Although these functions grow quickly for large n, their growth rate is slower compared to exponential functions when n approaches infinity.

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Taylor Polynomials

Limit Comparison of Growth Rates

To compare growth rates as n → ∞, we analyze the limit of the ratio of the two functions. If the limit is zero, the numerator grows slower; if infinite, it grows faster. This method helps determine which function dominates in the long run.

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Percorso guidato

04:16

Intro To Related Rates

Pratica correlata

Domanda del libro di testo

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) 3ᵏ / (k² + 1)

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Domanda del libro di testo

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.

67. ∑ (k = 1 to ∞) 3 / (k² + 5k + 4)

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Domanda del libro di testo

6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.

{1.00001ⁿ}

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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 / n)¹⁄ⁿ}

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Domanda del libro di testo

11–27. Alternating Series Test Determine whether the following series converge.

∑ (k = 0 to ∞) (−1)ᵏ / (2k + 1)

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