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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblema 10.8.25
Capitolo 10, Problema 10.8.25

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 1 to ∞) 1 / (√k × e^(√k))

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Identify the general term of the series: \(a_k = \frac{1}{\sqrt{k} \times e^{\sqrt{k}}}\).
Consider the behavior of the term \(a_k\) as \(k \to \infty\). Since \(e^{\sqrt{k}}\) grows very rapidly, the terms \(a_k\) approach zero.
To determine convergence, apply the Comparison Test by comparing \(a_k\) to a simpler series. Note that \(e^{\sqrt{k}}\) grows faster than any polynomial, so compare \(a_k\) to \(\frac{1}{e^{\sqrt{k}}}\).
Since \(\sum \frac{1}{e^{\sqrt{k}}}\) converges (because the terms decrease exponentially), and \(a_k < \frac{1}{e^{\sqrt{k}}}\) for all large \(k\), by the Comparison Test, the original series converges.
Alternatively, you can apply the Integral Test by considering the integral of the function \(f(x) = \frac{1}{\sqrt{x} e^{\sqrt{x}}}\) from 1 to infinity and showing that it converges, which implies the series converges.

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Convergence of Infinite Series

An infinite series converges if the sequence of its partial sums approaches a finite limit. Determining convergence involves analyzing the behavior of the terms as the index grows large, ensuring the sum does not diverge to infinity.
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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. If terms of the given series behave similarly to a convergent or divergent series, we can conclude the same about the original series, simplifying convergence analysis.
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Limit Comparison Test

Exponential and Root Functions in Series Terms

Understanding how exponential functions like e^(√k) grow faster than polynomial or root functions is crucial. This growth rate often dominates the denominator, causing terms to decrease rapidly, which influences the convergence of the series.
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Exponential Functions
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9–16. Divergence Test Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive.


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11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.

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21–42. Geometric series Evaluate each geometric series or state that it diverges.  


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48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

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