In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]

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Identify the general term of the series: \(a_n = \left( \frac{n+1}{n+2} \right)^n\).

To determine convergence or divergence, consider using the Root Test or the Ratio Test. The Root Test is often simpler for terms raised to the power \(n\).

Apply the Root Test by evaluating \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \left| \frac{n+1}{n+2} \right|\).

Simplify the limit: since \(\frac{n+1}{n+2} = 1 - \frac{1}{n+2}\), the limit approaches 1 as \(n\) approaches infinity.

Interpret the Root Test result: if the limit equals 1, the test is inconclusive, so consider using the Ratio Test or analyze the behavior of \(a_n\) directly to decide on convergence or divergence.

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주요 개념

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining whether a series converges means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding convergence is essential to analyze the behavior of the given series.

Limit of the General Term

A necessary condition for a series to converge is that its general term approaches zero as n approaches infinity. If the limit of the term does not equal zero, the series diverges. Evaluating this limit helps quickly assess the possibility of convergence.

Root Test for Series Convergence

The root test uses the nth root of the absolute value of the general term to determine convergence. If the limit of this root is less than one, the series converges absolutely; if greater than one, it diverges. This test is particularly useful for series with terms raised to the nth power.

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]