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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.5.59
Capitolo 10, Problema 10.5.59

40–62. Choose your test Use the test of your choice to determine whether the following series converge.
∑ (k = 1 to ∞) tan(1 / k)

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Identify the series given: \( \sum_{k=1}^{\infty} \tan\left(\frac{1}{k}\right) \). We want to determine if this series converges or diverges.
Recall that for large \( k \), \( \frac{1}{k} \) approaches 0, so consider the behavior of \( \tan(x) \) as \( x \to 0 \). Use the fact that \( \tan(x) \approx x \) when \( x \) is close to 0.
Approximate the terms of the series for large \( k \): \( \tan\left(\frac{1}{k}\right) \approx \frac{1}{k} \). This suggests the series behaves similarly to the harmonic series \( \sum \frac{1}{k} \).
Since the harmonic series \( \sum \frac{1}{k} \) diverges, use the Limit Comparison Test with \( a_k = \tan\left(\frac{1}{k}\right) \) and \( b_k = \frac{1}{k} \). Compute \( \lim_{k \to \infty} \frac{a_k}{b_k} \).
If the limit is a finite nonzero number, then both series either converge or diverge together. Since \( \sum \frac{1}{k} \) diverges, conclude the behavior of the original series accordingly.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Understanding convergence is essential to determine whether the sum of infinitely many terms results in a finite value or diverges to infinity or oscillates.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests compare the given series to a known benchmark series to determine convergence. The limit comparison test involves taking the limit of the ratio of terms from two series, helping to conclude convergence or divergence based on the behavior of the simpler series.
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Limit Comparison Test

Behavior of tan(1/k) for Large k

As k approaches infinity, 1/k approaches zero, and tan(1/k) behaves similarly to 1/k because tan(x) ~ x near zero. Recognizing this helps approximate the series terms and apply appropriate convergence tests by comparing to the harmonic series.
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Domanda del libro di testo

The first ten terms of the sequence {(1 + 1/10ⁿ)^10ⁿ}∞ ₙ₌₁ are rounded to 8 digits right of the decimal point (see table). Make a conjecture about the limit of the sequence.

n an

1 2.59374246

2 2.70481383

3 2.71692393

4 2.71814593

5 2.71826824

6 2.71828047

7 2.71828169

8 2.71828179

9 2.71828204

10 2.71828203

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9–15. Geometric sums Evaluate each geometric sum.


{Use of Tech}∑ k = 0 to 9(−3/4)ᵏ

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61–66. Sequences of partial sums For the following infinite series, find the first four terms of the sequence of partial sums. Then make a conjecture about the value of the infinite series or state that the series diverges.


4 + 0.9 + 0.09 + 0.009 + ⋯  

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45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞

 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.

aₙ = ⁿ² + n ;n = 1, 2, 3, …

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32–49. Choose your test Use the test of your choice to determine whether the following series converge absolutely, converge conditionally, or diverge.

∑ (from k = 1 to ∞) 2ᵏ k! / kᵏ

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54–69. Telescoping series

For the following telescoping series, find a formula for the nth term of the sequence of partial sums {Sₙ}. Then evaluate limₙ→∞ Sₙ to obtain the value of the series or state that the series diverges.


65. ∑ (k = 1 to ∞) (1 / √(k + 1) – 1 / √(k + 3))

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