Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}
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14. Sequences & Series
Sequences
1:29 minutesProblem 10.2.77
Textbook Question
Growth rates of sequences
Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.
{n¹⁰ / ln20n}
Verified step by step guidance1
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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Video duration:
1mKey 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 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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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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