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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 10.2.77
Capitolo 10, Problema 10.2.77

Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.


{n¹⁰ / ln20n}

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Identify the sequence given: \(a_n = \frac{n^{10}}{\ln(20n)}\).
Recall Theorem 10.6, which typically deals with growth rates of sequences involving polynomials and logarithmic functions. It states that polynomial functions grow faster than logarithmic functions as \(n \to \infty\).
Analyze the numerator and denominator separately: the numerator \(n^{10}\) grows very quickly (polynomial growth), while the denominator \(\ln(20n)\) grows slowly (logarithmic growth).
Since the numerator grows much faster than the denominator, the fraction \(\frac{n^{10}}{\ln(20n)}\) increases without bound as \(n\) becomes very large.
Conclude that the limit of the sequence \(a_n\) as \(n \to \infty\) is \(+\infty\), meaning the sequence diverges to infinity.

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Limits of Sequences

The limit of a sequence describes the value that the terms approach as the index goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges to infinity or does not settle.
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Introduction to Sequences

Growth Rates of Functions

Comparing growth rates involves analyzing how fast functions like polynomials, logarithms, and exponentials increase as their input grows. Polynomials grow faster than logarithmic functions, which is crucial for determining the behavior of sequences involving these terms.
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Intro To Related Rates

Theorem 10.6 (Comparison of Growth Rates)

Theorem 10.6 typically states that for large n, polynomial functions dominate logarithmic functions, meaning n^k grows faster than ln(n) for any positive integer k. This theorem helps conclude that sequences with polynomial numerators and logarithmic denominators tend to infinity.
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04:16
Intro To Related Rates
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