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

aₙ = (−1)ⁿ ⁿ√n

검증된 단계별 안내

Identify the given sequence: \(a_n = (-1)^n \sqrt{n}\), where \(\sqrt{n}\) denotes the square root of \(n\).

Recall that the term \((-1)^n\) causes the sequence to alternate signs between positive and negative values as \(n\) increases.

Analyze the behavior of the magnitude of the terms, which is \(\sqrt{n}\). As \(n\) approaches infinity, \(\sqrt{n}\) grows without bound (it increases without limit).

Consider the combined effect: since the magnitude \(\sqrt{n}\) grows larger and the sign alternates, the terms will oscillate between increasingly large positive and negative values.

Conclude that because the terms do not approach a single finite value, the sequence \(a_n\) does not converge and therefore diverges.

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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 n goes to infinity. If the terms get arbitrarily close to a specific number, the sequence converges; otherwise, it diverges. Understanding limits is essential to analyze the behavior of sequences like aₙ = (−1)ⁿ n^(1/n).

Nth Root and Its Limit

The nth root of n, expressed as n^(1/n), approaches 1 as n becomes very large. This is because the growth of n is tempered by the root, which grows slower, causing the expression to stabilize near 1. Recognizing this helps simplify the sequence's behavior.

Alternating Sequences

Sequences with terms multiplied by (−1)ⁿ alternate in sign between positive and negative values. This oscillation can affect convergence, as the sequence may not settle on a single value. Analyzing the impact of the alternating factor is crucial to determine if the sequence converges or diverges.

55–70. More sequencesFind the limit of the following sequences or determine that the sequence diverges.aₙ = (−1)ⁿ ⁿ√n