Are there any values of x for which ∑ (from n=1 to ∞) (1 / nˣ) converges? Give reasons for your answer.

검증된 단계별 안내

Recognize that the given series is the p-series (or more specifically, the Riemann zeta function series) defined as \(\sum_{n=1}^{\infty} \frac{1}{n^x}\), where \(x\) is a real number.

Recall the convergence criteria for p-series: the series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if and only if \(p > 1\) and diverges otherwise.

Apply this criterion to the given series by comparing \(x\) to 1. If \(x > 1\), the terms \(\frac{1}{n^x}\) decrease sufficiently fast for the series to converge.

If \(x \leq 1\), the terms do not decrease fast enough, and the series diverges. For example, when \(x=1\), the series becomes the harmonic series, which is known to diverge.

Therefore, the series \(\sum_{n=1}^{\infty} \frac{1}{n^x}\) converges if and only if \(x > 1\).

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주요 개념

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

Definition of the p-Series

A p-series is an infinite series of the form ∑ 1/n^p, where p is a real number. Its convergence depends on the value of p, serving as a fundamental example in series convergence tests.

Convergence Criteria for p-Series

The p-series ∑ 1/n^p converges if and only if p > 1. For p ≤ 1, the series diverges. This result is crucial for determining the convergence of series like ∑ 1/n^x by comparing the exponent x to 1.

Comparison Test for Series Convergence

The comparison test involves comparing a given series to a known benchmark series to determine convergence. If the terms of the given series are smaller than those of a convergent series, it also converges, aiding in analyzing series like ∑ 1/n^x.