13–52. Limits of sequences
Find the limit of the following sequences or determine that the sequence diverges.

{(1 / n)¹⁄ⁿ}

검증된 단계별 안내

Identify the given sequence: \(a_n = \left( \frac{1}{n} \right)^{\frac{1}{n}}\).

Rewrite the sequence using properties of exponents and logarithms to make the limit easier to analyze: \(a_n = e^{\ln \left( \left( \frac{1}{n} \right)^{\frac{1}{n}} \right)} = e^{\frac{1}{n} \ln \left( \frac{1}{n} \right)}\).

Simplify the exponent: \(\frac{1}{n} \ln \left( \frac{1}{n} \right) = \frac{1}{n} (-\ln n) = -\frac{\ln n}{n}\).

Analyze the limit of the exponent as \(n\) approaches infinity: \(\lim_{n \to \infty} -\frac{\ln n}{n}\). Recall that \(\ln n\) grows slower than \(n\), so this limit tends to zero.

Conclude the limit of the sequence by applying the limit to the exponential form: \(\lim_{n \to \infty} a_n = e^{\lim_{n \to \infty} -\frac{\ln n}{n}} = e^0\).

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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 is 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 to that limit; otherwise, it diverges.

Properties of Exponents and Roots

Understanding how to manipulate expressions with fractional exponents and roots is essential. For example, (1/n)^(1/n) can be rewritten using exponent rules to analyze its behavior as n increases.

Use of Logarithms in Limit Evaluation

Taking the natural logarithm of a sequence can simplify limit evaluation, especially for expressions involving exponents. By analyzing the limit of the logarithm, one can often find the original sequence's limit using continuity of the exponential function.

13–52. Limits of sequencesFind the limit of the following sequences or determine that the sequence diverges.{(1 / n)¹⁄ⁿ}