Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{(−1)ⁿ / 2ⁿ}
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14. Sequences & Series
Sequences
1:35 minutesProblem 10.2.63
Textbook Question
55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.
{tan⁻¹n / n}
Verified step by step guidance1
Identify the given sequence as \( a_n = \frac{\tan^{-1}(n)}{n} \), where \( \tan^{-1}(n) \) is the inverse tangent (arctangent) function.
Recall the behavior of the arctangent function as \( n \to \infty \): \( \tan^{-1}(n) \) approaches \( \frac{\pi}{2} \) because the arctangent of very large positive numbers tends to \( \frac{\pi}{2} \).
Rewrite the limit expression using this information: \( \lim_{n \to \infty} \frac{\tan^{-1}(n)}{n} = \lim_{n \to \infty} \frac{\frac{\pi}{2}}{n} \) approximately for large \( n \).
Since the denominator \( n \) grows without bound and the numerator approaches a constant \( \frac{\pi}{2} \), analyze the limit of a constant divided by an infinitely large number.
Conclude that the limit of the sequence is \( 0 \) because dividing a finite constant by an infinitely large number tends to zero.
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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 of the sequence approach as the index goes to infinity. Understanding how to evaluate limits helps determine whether a sequence converges to a finite number or diverges.
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Introduction to Sequences
Behavior of the Arctangent Function
The arctangent function, tan⁻¹(x), is continuous and bounded, approaching π/2 as x approaches infinity. Knowing this asymptotic behavior is crucial for analyzing sequences involving tan⁻¹(n) as n grows large.
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Growth Rates and Dominant Terms
Comparing the growth rates of numerator and denominator helps determine the limit of a sequence. Since n grows without bound and tan⁻¹(n) approaches a constant, understanding which term dominates is key to finding the sequence's limit.
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