Ch. 10 - Infinite Sequences and Series

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 10 - Infinite Sequences and Series

Problema 10.6.94

Capitolo 10, Problema 10.6.94

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

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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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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.

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06:52

Convergence of an Infinite Series

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.

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09:25

Direct Comparison Test

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.

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P-Series and Harmonic Series

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Absolute and Conditional Convergence

Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.

∑ (from n = 1 to ∞) [(-1)ⁿ⁺¹ (ⁿ√10)]

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Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 0 to ∞) e^(−2n)

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In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 0 to ∞) [((n + 1) / (n + 2))ⁿ]

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Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

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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 ∞) (n / (3n + 1))ⁿ

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Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

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