Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 11 - Power Series

Problema 11.RE.20

Capitolo 11, Problema 11.RE.20

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞
Σ (x - 1)ᵏ/(k5ᵏ)
k = 1

Guida verificata passo dopo passo

Identify the general term of the power series: \(a_k = \frac{(x - 1)^k}{k 5^k}\).

Apply the Ratio Test, which involves computing the limit \(L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|\). Substitute \(a_k\) and \(a_{k+1}\) into this expression.

Simplify the ratio inside the limit: \(\left| \frac{(x - 1)^{k+1}}{(k+1) 5^{k+1}} \cdot \frac{k 5^k}{(x - 1)^k} \right| = \left| \frac{x - 1}{5} \cdot \frac{k}{k+1} \right|\).

Evaluate the limit as \(k\) approaches infinity: \(L = \left| \frac{x - 1}{5} \right| \cdot \lim_{k \to \infty} \frac{k}{k+1} = \left| \frac{x - 1}{5} \right|\).

Set the limit \(L\) less than 1 to find the radius of convergence: \(\left| \frac{x - 1}{5} \right| < 1\). Solve this inequality to find the interval of convergence before testing the endpoints.

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Radius of Convergence

The radius of convergence of a power series is the distance from the center of the series within which the series converges absolutely. It defines an interval on the x-axis where the series behaves well. Finding this radius helps determine where the series can be used to represent a function accurately.

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07:36

Radius of Convergence

Ratio Test and Root Test

The Ratio Test and Root Test are methods to determine the convergence of infinite series. The Ratio Test examines the limit of the ratio of successive terms, while the Root Test looks at the nth root of the absolute value of terms. Both tests help find the radius of convergence for power series by analyzing term behavior as k approaches infinity.

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Percorso guidato

07:15

Root Test

Interval of Convergence and Endpoint Testing

The interval of convergence is the full set of x-values for which a power series converges. After finding the radius, endpoints must be tested separately because convergence at these points is not guaranteed. Testing endpoints involves substituting them into the series and checking for convergence using appropriate tests.

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Percorso guidato

07:51

Choosing a Convergence Test

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Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞

Σ x⁴ᵏ/k²

k = 1

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Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)

ƒ(x) = ln (1 - x); bound R₃(x), for |x| < 1/2

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x +x³/3 +x⁵/5 +x⁷/7 + ...

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∞

Σ (x/9)³ᵏ

k = 0

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Domanda del libro di testo

Taylor series