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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.2.89
Capitolo 10, Problema 10.2.89

Make up an infinite series of nonzero terms whose sum is
b. −3

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Recall that an infinite series is the sum of infinitely many terms, and for the series to have a finite sum, it must be convergent.
One common type of infinite series with a known sum is a geometric series, which has the form \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\), where \(|r| < 1\).
To create a series that sums to \(b - 3\), choose the first term \(a\) and common ratio \(r\) such that \(\frac{a}{1-r} = b - 3\).
For example, pick a value for \(r\) with \(|r| < 1\), then solve for \(a = (b - 3)(1 - r)\) to ensure the sum matches \(b - 3\).
Write the infinite series explicitly as \(\sum_{n=0}^{\infty} a r^n\), substituting the values of \(a\) and \(r\) you chose, which will be a series of nonzero terms summing to \(b - 3\).

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. For the series to have a finite sum, it must converge, meaning its partial sums approach a specific value. Understanding convergence criteria is essential to construct a series that sums to a given number.
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06:52
Convergence of an Infinite Series

Geometric Series

A geometric series has terms that multiply by a constant ratio each time. If the absolute value of the ratio is less than one, the series converges to a sum given by a/(1 - r), where a is the first term and r is the ratio. This formula helps create series with a desired sum.
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06:00
Geometric Series

Constructing Series with a Given Sum

To create an infinite series summing to a specific value, choose terms that satisfy convergence and sum conditions. For example, adjusting the first term and ratio in a geometric series allows the sum to equal any real number, including negative values like −3.
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06:45
Intro to Series: Partial Sums
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Determining Convergence or Divergence

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.)

∑ (from n=1 to ∞) eⁿ / (1 + e²ⁿ)

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Finding Terms of a Sequence

Each of Exercises 1–6 gives a formula for the nth term aₙ of a sequence {aₙ}. Find the values of a₁, a₂, a₃, and a₄.

aₙ = 1 / n!

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In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.

1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

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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 = 1 to ∞) [3ⁿ / n³]

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Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1

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Limit Comparison Test

In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

∑ (from n=2 to ∞) 1 / ln n

(Hint: Limit Comparison with ∑ (from n=2 to ∞) (1/n))

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