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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.20
Chapter 11, Problem 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

Verified step by step guidance
1
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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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 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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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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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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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 x⁴ᵏ/k²

k = 1

1
views
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

ƒ(x) = eˣ, a = 0; e-0.08


b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.

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.

1
views
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/9)³ᵏ

k = 0

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.