55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.

{tan⁻¹n / n}

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.{tan⁻¹n / n}