40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

검증된 단계별 안내

Identify the given series: \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \) where \( p > 0 \). This is a p-series, a common type of series in calculus.

Recall the p-series test: A series of the form \( \sum_{k=1}^{\infty} \frac{1}{k^q} \) converges if and only if \( q > 1 \).

Compare the exponent in the denominator of the given series, which is \( 1 + p \), with 1. Since \( p > 0 \), then \( 1 + p > 1 \).

Apply the p-series test: Because \( 1 + p > 1 \), the series \( \sum_{k=1}^{\infty} \frac{1}{k^{1+p}} \) converges.

Conclude that the series converges for all \( p > 0 \) based on the p-series convergence criterion.

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주요 개념

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

p-Series Test

The p-series test determines the convergence of series of the form ∑ 1/k^p. Such a series converges if and only if p > 1, and diverges otherwise. This test is fundamental for analyzing series with terms involving powers of k.

Comparison Test

The comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence. 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.

Convergence of Infinite Series

Understanding convergence means determining whether the sum of infinitely many terms approaches a finite limit. Tests like the p-series test and comparison test help decide if the infinite sum converges or diverges, which is crucial for analyzing series behavior.

40–62. Choose your test Use the test of your choice to determine whether the following series converge.∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0