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 / (k + 10))ᵏ

검증된 단계별 안내

First, identify the general term of the series: \(a_k = \left( \frac{k}{k + 10} \right)^k\).

Apply the Divergence Test by finding the limit of \(a_k\) as \(k\) approaches infinity: compute \(\lim_{k \to \infty} \left( \frac{k}{k + 10} \right)^k\).

Rewrite the term inside the limit to a form that is easier to analyze: \(\left( \frac{k}{k + 10} \right)^k = \left( 1 - \frac{10}{k + 10} \right)^k\).

Recognize that this limit resembles the form \(\lim_{n \to \infty} \left( 1 - \frac{c}{n} \right)^n = e^{-c}\) for some constant \(c\), and use this to evaluate the limit.

If the limit of \(a_k\) is not zero, conclude by the Divergence Test that the series diverges; if the limit is zero, consider applying other tests such as the Root Test or Ratio Test for further analysis.

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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 cannot converge, but if the limit is zero, the test is inconclusive.

Integral Test

The Integral Test relates the convergence of a series to the convergence of an improper integral. If a function f(k) is positive, continuous, and decreasing for k ≥ 1, then the series ∑f(k) and the integral ∫f(x)dx from 1 to infinity either both converge or both diverge.

p-Series Test

A p-series is a series of the form ∑ 1/k^p. It converges if and only if p > 1 and diverges otherwise. This test helps quickly determine convergence for series resembling p-series or can be used as a comparison benchmark.

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 / (k + 10))ᵏ