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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 10.5.46
Capitolo 10, Problema 10.5.46

Determining Convergence or Divergence
In Exercises 17–46, use any method to determine whether the series converges or diverges. Give reasons for your answer.
∑(from n=3 to ∞) [2n² / n²ⁿ]

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First, write down the general term of the series: \(a_n = \frac{2n^2}{n^{2n}}\) for \(n \geq 3\).
Observe the form of \(a_n\) to decide which convergence test is appropriate. Since the term involves \(n\) raised to a power that depends on \(n\), consider using the Root Test or the Ratio Test.
Apply the Root Test by computing \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\frac{2n^2}{n^{2n}}}\).
Simplify the expression inside the limit: \(\sqrt[n]{2n^2} = (2n^2)^{1/n}\) and \(\sqrt[n]{n^{2n}} = n^{2n/n} = n^2\). So the limit becomes \(\lim_{n \to \infty} \frac{(2n^2)^{1/n}}{n^2}\).
Evaluate the limit: as \(n \to \infty\), \((2n^2)^{1/n} \to 1\) because the \(n\)th root of any polynomial grows slowly, while \(n^2\) in the denominator grows without bound. Therefore, the limit is \(0\), which is less than \(1\), indicating that the series converges by the Root Test.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining convergence means checking if the sum approaches a finite limit as the number of terms grows. If the series converges, its partial sums approach a specific value; if it diverges, the sums grow without bound or oscillate.
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Convergence of an Infinite Series

Comparison and Limit Comparison Tests

These tests help determine convergence by comparing the given series to a known benchmark series. The Comparison Test uses inequalities to relate terms, while the Limit Comparison Test uses the limit of the ratio of terms. Both are useful when terms involve polynomials and exponentials.
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Exponential Growth vs Polynomial Growth

Exponential functions grow faster than any polynomial function as n approaches infinity. In series terms, if the denominator grows exponentially and the numerator polynomially, the terms tend to zero rapidly, often implying convergence. Recognizing this helps in choosing the right test.
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Exponential Growth & Decay
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∑ (from n = 2 to ∞) [(3n)! / (n!)³]

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∑ (from n=1 to ∞) (1 + cos n) / n²

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∑ (from n=1 to ∞) 1 / (1 + 2² + 3² + ⋯ + n²)

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∑(from n=1 to ∞) [n! / (2n + 1)!]