Determining Convergence or Divergence
Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)
∑ (from n=1 to ∞) (1 + 1/n)ⁿ

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Identify the general term of the series: \(a_n = \left(1 + \frac{1}{n}\right)^n\).

Recall that the series is \(\sum_{n=1}^\infty a_n\), and to determine convergence, first check the behavior of the terms \(a_n\) as \(n\) approaches infinity.

Evaluate the limit \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\). Recognize this limit as the definition of the mathematical constant \(e\).

Since \(\lim_{n \to \infty} a_n = e \neq 0\), apply the Divergence Test (also called the Test for Divergence), which states that if the limit of the terms does not equal zero, the series diverges.

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

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주요 개념

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Convergence means the series approaches a finite limit as the number of terms grows, while divergence means it does not. Understanding whether a series converges or diverges is fundamental to analyzing its behavior.

Limit of the General Term

For a series ∑a_n to converge, the terms a_n must approach zero as n approaches infinity. If the limit of a_n is not zero, the series diverges by the Test for Divergence. This is a quick initial check before applying more complex tests.

Comparison and Root Tests

The Comparison Test compares a series to a known benchmark series to determine convergence. The Root Test uses the nth root of the absolute value of terms to assess convergence: if this limit is less than 1, the series converges; if greater than 1, it diverges. These tests help analyze series with terms involving powers or roots.

Determining Convergence or DivergenceWhich of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)∑ (from n=1 to ∞) (1 + 1/n)ⁿ

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Infinite Series and Convergence

Limit of the General Term

Comparison and Root Tests