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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblema 10.PE.68
Capitolo 10, Problema 10.PE.68

Maclaurin Series
Find Taylor series at x = 0 for the functions in Exercises 63–70.
cos (x³/√5)

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1
Recall that the Maclaurin series is a Taylor series expansion of a function at \(x = 0\). It can be written as \(f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n\).
Identify the function to expand: \(f(x) = \cos\left(\frac{x^3}{\sqrt{5}}\right)\). Notice that the argument of cosine is \(\frac{x^3}{\sqrt{5}}\).
Recall the Maclaurin series expansion for \(\cos z\) is \(\cos z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n}}{(2n)!}\), where \(z\) is any expression.
Substitute \(z = \frac{x^3}{\sqrt{5}}\) into the cosine series to get \(\cos\left(\frac{x^3}{\sqrt{5}}\right) = \sum_{n=0}^{\infty} (-1)^n \frac{\left(\frac{x^3}{\sqrt{5}}\right)^{2n}}{(2n)!}\).
Simplify the powers and write the series explicitly as \(\sum_{n=0}^{\infty} (-1)^n \frac{x^{6n}}{(2n)! (\sqrt{5})^{2n}}\). This is the Maclaurin series for the given function.

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Maclaurin Series

A Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of its derivatives at zero, multiplied by powers of x and divided by factorial terms. This series helps approximate functions near zero.
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Convergence of Taylor & Maclaurin Series

Taylor Series Expansion

The Taylor series expresses a function as an infinite sum of terms calculated from the function's derivatives at a specific point. For a function f(x), the series at x = a is given by f(a) plus derivatives evaluated at a, scaled by powers of (x - a). When a = 0, it becomes the Maclaurin series.
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Taylor Series

Series Expansion of Composite Functions

When dealing with functions like cos(x³/√5), the series expansion involves substituting the inner function into the known series of the outer function. This requires understanding how to replace variables in standard series and simplify powers and coefficients accordingly.
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Percorso guidato
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Geometric Series
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Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;


ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series


a₁/1 + a₂/2 + a₃/3 + …


diverges.

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Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.


∑ (from n = 1 to ∞) (x + 4)ⁿ/(n3ⁿ)

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Convergent Series

Find the sums of the series in Exercises 19–24.


∑ (from n = 2 to ∞) -2/[n(n+1)]

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Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) xⁿ/nⁿ

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Power Series

In Exercises 47–56, (a) find the series’ radius and interval of convergence. Then identify the values of x for which the series converges (b) absolutely and (c) conditionally.

∑ (from n = 1 to ∞) (csch n)xⁿ

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Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.


aₙ = 1 + (0.9)ⁿ

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