23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.
∑ (k = 1 to ∞) k^(1/k)

검증된 단계별 안내

First, identify the general term of the series: \(a_k = k^{1/k}\).

Apply the Divergence Test by finding the limit of \(a_k\) as \(k\) approaches infinity: compute \(\lim_{k \to \infty} k^{1/k}\).

Rewrite the term inside the limit using exponentials and logarithms: \(k^{1/k} = e^{(1/k) \ln(k)}\).

Evaluate the limit of the exponent: \(\lim_{k \to \infty} \frac{\ln(k)}{k}\), which approaches 0, so the limit of \(a_k\) is \(e^0 = 1\).

Since the limit of \(a_k\) is 1 (not zero), by the Divergence Test, the series \(\sum_{k=1}^\infty k^{1/k}\) diverges.

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주요 개념

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

Divergence Test

The Divergence Test states that if the limit of the terms of a series does not approach zero as k approaches infinity, then the series diverges. It is a quick initial check to determine if a series can possibly converge.

Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral of a related function. If the integral of the function from 1 to infinity converges, then the series converges; if the integral diverges, so does the series.

p-series Test

The p-series Test applies to series of the form ∑ 1/k^p. Such a series converges if and only if p > 1, and diverges otherwise. It is useful for comparing or identifying the behavior of series with terms involving powers of k.

23–38. Divergence, Integral, and p-series Tests Use the Divergence Test, the Integral Test, or the p-series test to determine whether the following series converge.∑ (k = 1 to ∞) k^(1/k)