Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) 1 / (2√n + ³√n)

검증된 단계별 안내

Identify the general term of the series: \(a_n = \frac{1}{2\sqrt{n} + \sqrt[3]{n}}\).

Analyze the behavior of the denominator for large \(n\). Since \(\sqrt{n} = n^{1/2}\) and \(\sqrt[3]{n} = n^{1/3}\), the term \(2\sqrt{n}\) grows faster than \(\sqrt[3]{n}\) as \(n\) increases.

Approximate the general term for large \(n\) by focusing on the dominant term in the denominator: \(a_n \approx \frac{1}{2n^{1/2}}\).

Compare the series to a known benchmark series, such as the \(p\)-series \(\sum \frac{1}{n^p}\), where convergence depends on \(p\). Here, the comparison is with \(\sum \frac{1}{n^{1/2}}\).

Recall that the \(p\)-series \(\sum \frac{1}{n^p}\) converges if and only if \(p > 1\). Since \(1/2 < 1\), the comparison series diverges, so by the Comparison Test, the original series also diverges.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

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. If the sum does not approach a finite value, the series diverges.

Comparison Test

The Comparison Test helps determine convergence by comparing the given series to a known benchmark series. 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. This test is useful when terms are positive and can be bounded.

Asymptotic Behavior of Terms

Analyzing the dominant behavior of terms for large n helps simplify complex expressions. For example, in 1/(2√n + ³√n), the term with the slower decay rate (smaller exponent) dominates. Understanding which term controls the behavior is key to applying convergence tests effectively.

Determining Convergence or DivergenceWhich of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.∑ (from n=1 to ∞) 1 / (2√n + ³√n)