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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.35
Capitolo 10, Problema 10.8.35

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 2⁹k / kᵏ

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First, write down the general term of the series: \(a_k = \frac{2^{9k}}{k^k}\).
To determine convergence, consider applying the Root Test, which is useful for series with terms raised to the power of \(k\). The Root Test uses the limit \(L = \lim_{k \to \infty} \sqrt[k]{|a_k|}\).
Calculate \(\sqrt[k]{|a_k|} = \sqrt[k]{\frac{2^{9k}}{k^k}} = \frac{2^9}{k}\), since \(\sqrt[k]{2^{9k}} = 2^9\) and \(\sqrt[k]{k^k} = k\).
Evaluate the limit \(L = \lim_{k \to \infty} \frac{2^9}{k}\). As \(k\) approaches infinity, \(\frac{2^9}{k}\) approaches 0.
Since \(L = 0 < 1\), by the Root Test, the series \(\sum_{k=1}^\infty \frac{2^{9k}}{k^k}\) converges absolutely.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.
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Convergence of an Infinite Series

Root Test for Convergence

The Root Test involves taking the k-th root of the absolute value of the k-th term and examining its limit as k approaches infinity. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. This test is particularly useful for series with terms raised to the k-th power.
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07:15
Root Test

Exponential and Factorial Growth Rates

Comparing growth rates of functions like exponentials and powers is crucial in convergence tests. In the series ∑ (2^(9k) / k^k), the denominator grows faster than any exponential due to k^k, which tends to infinity much faster, suggesting the terms approach zero rapidly, influencing convergence.
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Factorials
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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


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43–44. Periodic doses

Suppose you take a dose of m mg of a particular medication once per day. Assume f equals the fraction of the medication that remains in your blood one day later. Just after taking another dose of medication on the second day, the amount of medication in your blood equals the sum of the second dose and the fraction of the first dose remaining in your blood, which is m + mf. Continuing in this fashion, the amount of medication in your blood just after your nth dose is


Aₙ = m + mf + ⋯ + mfⁿ⁻¹.


For the given values of f and m, calculate A₅, A₁₀, A₃₀, and lim (n → ∞) Aₙ. Interpret the meaning of the limit lim (n → ∞) Aₙ.


43.f = 0.25,m = 200 mg

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9–30. The Ratio and Root Tests Use the Ratio Test or the Root Test to determine whether the following series converge absolutely or diverge.

∑ (from k = 1 to ∞) ((-7)ᵏ / k²)

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What test is advisable if a series involves a factorial term?

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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.


∑ (from k = 1 to ∞)(1 / √(k + 2) – 1 / √k)

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Evaluate 1000!/998! without a calculator.

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