In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 1 to ∞) tan(n^(1/n))

검증된 단계별 안내

First, analyze the general term of the series: \(a_n = \tan\left(n^{\frac{1}{n}}\right)\). To determine convergence or divergence, we need to understand the behavior of \(a_n\) as \(n\) approaches infinity.

Evaluate the limit of the inside of the tangent function: \(\lim_{n \to \infty} n^{\frac{1}{n}}\). Recall that \(n^{\frac{1}{n}} = e^{\frac{\ln n}{n}}\). Since \(\frac{\ln n}{n} \to 0\) as \(n \to \infty\), this limit approaches \(e^0 = 1\).

Use the limit found to find \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \tan\left(n^{\frac{1}{n}}\right) = \tan(1)\). Since \(\tan(1)\) is a finite nonzero number, the terms \(a_n\) do not approach zero.

Recall the Divergence Test (also called the nth-term test for divergence): if \(\lim_{n \to \infty} a_n \neq 0\), then the series \(\sum a_n\) diverges. Since \(\lim_{n \to \infty} a_n = \tan(1) \neq 0\), the series diverges.

Therefore, conclude that the series \(\sum_{n=1}^\infty \tan\left(n^{\frac{1}{n}}\right)\) diverges by the Divergence Test.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Behavior of the General Term in a Series

To determine convergence, first analyze the limit of the general term as n approaches infinity. If the term does not approach zero, the series diverges by the Test for Divergence. Understanding the behavior of tan(n^(1/n)) as n grows large is crucial here.

Limit Comparison and Standard Convergence Tests

Various tests like the Comparison Test, Limit Comparison Test, or the Divergence Test help determine series convergence. Choosing an appropriate test depends on the form of the terms; for example, comparing tan(n^(1/n)) to a simpler function can clarify convergence behavior.

Properties and Limits of the Function n^(1/n)

The expression n^(1/n) approaches 1 as n approaches infinity. Understanding this limit helps simplify tan(n^(1/n)) to approximately tan(1) for large n, which informs whether the terms tend to zero or not, a key step in analyzing series convergence.

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.∑ (from n = 1 to ∞) tan(n^(1/n))