11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from j = 1 to ∞) 5 / (j² + 4)

검증된 단계별 안내

Identify the given series: \( \sum_{j=1}^{\infty} \frac{5}{j^{2} + 4} \). We want to determine if this series converges or diverges.

Recognize that the terms \( \frac{5}{j^{2} + 4} \) are positive and decrease as \( j \) increases, since the denominator grows quadratically.

Compare the given series to a known benchmark series. Notice that \( \frac{5}{j^{2} + 4} < \frac{5}{j^{2}} \) for all \( j \geq 1 \). The series \( \sum_{j=1}^{\infty} \frac{5}{j^{2}} \) is a constant multiple of the p-series \( \sum \frac{1}{j^{2}} \) with \( p = 2 > 1 \), which is known to converge.

Apply the Comparison Test: since \( \sum \frac{5}{j^{2}} \) converges and \( \frac{5}{j^{2} + 4} \leq \frac{5}{j^{2}} \), the original series \( \sum \frac{5}{j^{2} + 4} \) also converges.

Conclude that the series converges by the Comparison Test, justifying the answer based on the behavior of the terms and the known convergence of the p-series.

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.

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.

Comparison Test

The Comparison Test involves comparing a given series to a second series whose convergence behavior is known. If the terms of the given series are smaller than those of a convergent series, it also converges; if larger than a divergent series, it diverges.

p-Series Test

A p-series is of the form ∑ 1/n^p, which converges if p > 1 and diverges otherwise. Recognizing that the given series resembles a p-series helps in applying this test to determine convergence by comparing the denominator's growth rate.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from j = 1 to ∞) 5 / (j² + 4)