Limit Comparison Test
In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.
∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))

검증된 단계별 안내

Identify the given series: \( \sum_{n=1}^{\infty} \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \). We want to determine if it converges or diverges using the Limit Comparison Test.

Choose a comparison series \( b_n \) that resembles the behavior of \( a_n = \frac{n(n + 1)}{(n^2 + 1)(n - 1)} \) for large \( n \). Simplify the dominant terms: numerator behaves like \( n^2 \), denominator behaves like \( n^3 \), so \( a_n \) behaves like \( \frac{n^2}{n^3} = \frac{1}{n} \). Thus, choose \( b_n = \frac{1}{n} \).

Compute the limit \( L = \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\frac{n(n + 1)}{(n^2 + 1)(n - 1)}}{\frac{1}{n}} = \lim_{n \to \infty} \frac{n(n + 1) \cdot n}{(n^2 + 1)(n - 1)} \).

Simplify the expression inside the limit and evaluate \( L \). If \( L \) is a finite positive number (i.e., \( 0 < L < \infty \)), then \( a_n \) and \( b_n \) have the same behavior regarding convergence.

Since \( \sum \frac{1}{n} \) (the harmonic series) diverges, use the result of the Limit Comparison Test to conclude whether the original series converges or diverges.

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.

Limit Comparison Test

The Limit Comparison Test is used to determine the convergence or divergence of an infinite series by comparing it to a second series with known behavior. It involves taking the limit of the ratio of the nth terms of the two series. If the limit is a positive finite number, both series either converge or diverge together.

Behavior of Rational Functions for Large n

When analyzing series with rational expressions, it is important to understand the dominant terms as n approaches infinity. Simplifying the highest degree terms in numerator and denominator helps approximate the general term's behavior, which is crucial for choosing an appropriate comparison series.

Convergence of p-Series

A p-series is a series of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing or approximating a given series to a p-series helps in applying the Limit Comparison Test effectively, as p-series have well-known convergence properties.

Limit Comparison TestIn Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))