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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.RE.18
Chapter 11, Problem 11.RE.18

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

Verified step by step guidance
1
Identify the general term of the power series: \(a_k = \frac{x^{4k}}{k^2}\).
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 the ratio: \(\left| \frac{a_{k+1}}{a_k} \right| = \left| \frac{x^{4(k+1)}}{(k+1)^2} \cdot \frac{k^2}{x^{4k}} \right| = \left| x^4 \right| \cdot \frac{k^2}{(k+1)^2}\).
Evaluate the limit as \(k\) approaches infinity: \(L = |x|^4 \cdot \lim_{k \to \infty} \frac{k^2}{(k+1)^2} = |x|^4\).
Use the Ratio Test criterion for convergence: the series converges if \(L < 1\), so \(|x|^4 < 1\), which implies \(|x| < 1\). This gives the radius of convergence \(R = 1\). Next, test the endpoints \(x = -1\) and \(x = 1\) by substituting into the original series and checking for convergence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 is found by analyzing the limit behavior of the series' terms, often using tests like the Ratio or Root Test. This radius defines an interval on the real line where the series converges.
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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 absolute value of consecutive term ratios, 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 evaluating limits involving the variable.
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Root Test

Interval of Convergence and Endpoint Testing

The interval of convergence is the set of all x-values for which a power series converges. After finding the radius of convergence, 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 convergence tests.
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Choosing a Convergence Test
Related Practice
Textbook Question

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

Textbook Question

Limits by power series Use Taylor series to evaluate the following limits.


lim ₙ → 0 (x²/2 - 1 + cos x)/x⁴

1
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Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series.


sinh (-1)

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Textbook Question

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) = eˣ; bound R₃(x), for |x| < 1

Textbook Question

A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.

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Textbook Question

ƒ(x) = eˣ, a = 0; e-0.08


a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point a.