Does the series
∑ (from n=1 to ∞) (1/n − 1/n²)
converge or diverge? Justify your answer.

검증된 단계별 안내

Rewrite the given series to understand its general term: the series is \( \sum_{n=1}^{\infty} \left( \frac{1}{n} - \frac{1}{n^2} \right) \).

Split the series into two separate series using the linearity of summation: \( \sum_{n=1}^{\infty} \frac{1}{n} - \sum_{n=1}^{\infty} \frac{1}{n^2} \).

Analyze the convergence of each series individually: \( \sum_{n=1}^{\infty} \frac{1}{n} \) is the harmonic series, which is known to diverge, and \( \sum_{n=1}^{\infty} \frac{1}{n^2} \) is a p-series with \( p=2 > 1 \), which converges.

Since the first series diverges and the second converges, consider the behavior of their difference. The divergence of the harmonic series dominates, so the original series behaves like \( \sum \frac{1}{n} \) for large \( n \).

Conclude that the original series diverges because subtracting a convergent series from a divergent one does not make it converge.

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주요 개념

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Series and Convergence

A series is the sum of the terms of a sequence. To determine if a series converges, we check if the sum approaches a finite limit as the number of terms grows indefinitely. If the partial sums do not approach a finite value, the series diverges.

Comparison Test for Series

The comparison test helps determine convergence by comparing a given series to a known benchmark series. If the terms of the given series are smaller than those of a convergent series, it converges; if larger than a divergent series, it diverges.

Behavior of p-Series

A p-series has the form ∑ 1/n^p. It converges if p > 1 and diverges if p ≤ 1. Recognizing parts of a series as p-series helps in analyzing convergence by applying this well-known criterion.

Does the series∑ (from n=1 to ∞) (1/n − 1/n²)converge or diverge? Justify your answer.