b.Does the series ∑ (from k = 1 to ∞) k/(k + 1) converge? Why or why not?

검증된 단계별 안내

Identify the general term of the series: \(a_k = \frac{k}{k+1}\).

Examine the behavior of the terms \(a_k\) as \(k\) approaches infinity by finding \(\lim_{k \to \infty} a_k\).

Calculate the limit: \(\lim_{k \to \infty} \frac{k}{k+1} = \lim_{k \to \infty} \frac{k}{k(1 + \frac{1}{k})} = \lim_{k \to \infty} \frac{1}{1 + \frac{1}{k}}\).

Since \(\lim_{k \to \infty} a_k = 1 \neq 0\), recall the necessary condition for series convergence: if the terms do not approach zero, the series cannot converge.

Conclude that because the terms do not approach zero, the series \(\sum_{k=1}^\infty \frac{k}{k+1}\) diverges.

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주요 개념

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

Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Convergence means the series approaches a finite limit as more terms are added. Determining convergence involves analyzing the behavior of the partial sums or applying convergence tests.

Term Test for Divergence

If the terms of a series do not approach zero as k approaches infinity, the series cannot converge. This is a quick initial test to rule out convergence by examining the limit of the general term.

Behavior of the General Term k/(k+1)

The term k/(k+1) simplifies to 1 - 1/(k+1), which approaches 1 as k grows large. Since the terms do not approach zero, the series ∑ k/(k+1) diverges by the term test.