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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.R.17
Chapter 10, Problem 10.R.17

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.
aₙ = 8ⁿ / n!

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Identify the sequence given: \(a_n = \frac{8^n}{n!}\), where \(n!\) denotes the factorial of \(n\).
Recall that factorial \(n!\) grows much faster than any exponential function \$8^n\( as \)n$ becomes very large.
To analyze the limit \(\lim_{n \to \infty} \frac{8^n}{n!}\), consider the behavior of the terms in the denominator compared to the numerator as \(n\) increases.
Use the ratio test for sequences by examining the ratio \(\frac{a_{n+1}}{a_n} = \frac{8^{n+1}/(n+1)!}{8^n/n!} = \frac{8}{n+1}\), which approaches 0 as \(n \to \infty\).
Since the ratio approaches 0, the terms \(a_n\) get smaller and smaller, indicating that the limit of the sequence is 0.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits of Sequences

The limit of a sequence describes the value that the terms of the sequence approach as the index n becomes very large. If the terms get arbitrarily close to a specific number, the sequence converges to that limit; otherwise, it diverges.
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Introduction to Sequences

Factorials and Growth Rates

Factorials (n!) grow much faster than exponential functions like 8ⁿ as n increases. Understanding this difference in growth rates helps determine the behavior of sequences involving factorials and exponentials.
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Factorials

Ratio Test for Sequences

The ratio test compares the ratio of consecutive terms aₙ₊₁ / aₙ to analyze the limit of a sequence. If this ratio approaches a value less than 1, the sequence tends to zero, aiding in evaluating limits involving factorials and exponentials.
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Ratio Test
Related Practice
Textbook Question

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

Textbook Question

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.


59. ∑ (k = –3 to ∞) 4 / ((4k – 3)(4k + 1))

Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 2 to ∞) 1 / (klnk)

Textbook Question

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

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Textbook Question

b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

Textbook Question

77–87. Absolute or conditional convergence

Determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞)(−1)ᵏ⁺¹(k² + 4) / (2k² + 1)