48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) k / √(k² + 1)

검증된 단계별 안내

First, write down the general term of the series: \(a_k = \frac{k}{\sqrt{k^2 + 1}}\).

To analyze convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Simplify the expression inside the square root to understand the dominant terms.

Divide numerator and denominator by \(k\) to rewrite \(a_k\) as \(\frac{k}{\sqrt{k^2 + 1}} = \frac{k}{k \sqrt{1 + \frac{1}{k^2}}} = \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\).

Evaluate the limit \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \frac{1}{\sqrt{1 + \frac{1}{k^2}}}\) to determine if the terms approach zero, which is necessary for convergence.

Since the terms do not approach zero, conclude that the series diverges by the Test for Divergence (also known as the nth-term 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.

Convergence and Divergence of Infinite Series

An infinite series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the sum does not approach a finite value, the series diverges. Understanding this distinction is fundamental to analyzing series behavior.

Limit Comparison Test

The Limit Comparison Test compares a given series with a known benchmark series by examining the limit of their term ratios. If the limit is a positive finite number, both series either converge or diverge together. This test is useful when terms resemble a simpler series.

Behavior of Terms and nth-Term Test for Divergence

The nth-term test states that if the limit of the terms of a series does not approach zero, the series must diverge. Analyzing the behavior of the general term, especially for large n, helps quickly identify divergence without further tests.

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.∑ (k = 1 to ∞) k / √(k² + 1)