11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ

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First, identify the general term of the series: \(a_k = \left(1 + \frac{1}{2k}\right)^k\).

To determine convergence, consider the behavior of \(a_k\) as \(k\) approaches infinity. Calculate the limit \(\lim_{k \to \infty} a_k = \lim_{k \to \infty} \left(1 + \frac{1}{2k}\right)^k\).

Recognize that this limit resembles the form of the exponential function \(e^x\), where \(\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n = e^x\). Here, \(x = \frac{1}{2}\), so the limit becomes \(e^{1/2}\).

Since the limit of the terms \(a_k\) is \(e^{1/2}\), which is a positive number not equal to zero, apply the Divergence Test (also called the Test for Divergence), which states that if \(\lim_{k \to \infty} a_k \neq 0\), then the series \(\sum a_k\) diverges.

Conclude that because the terms do not approach zero, the series \(\sum_{k=1}^\infty \left(1 + \frac{1}{2k}\right)^k\) diverges.

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주요 개념

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

Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.

Root Test for Convergence

The root test evaluates the limit of the k-th root of the absolute value of the terms. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges; if equal to 1, the test is inconclusive. This test is particularly useful for series with terms raised to the k-th power.

Behavior of Terms in the Series

Analyzing the general term (1 + 1/(2k))^k involves understanding limits and growth rates. Recognizing how the term behaves as k approaches infinity helps in applying convergence tests and deciding if terms approach zero, a necessary condition for convergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 1 to ∞)(1 + 1 / (2k))ᵏ