Suppose a power series converges if |x−3|\u003c4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.

Identify the center of the power series from the given inequality. Here, the series converges when $$|x - 3| \u003c 4$$, so the center is at $$x = 3. $$Recall that the radius of convergence $$R is $$the distance from the center to the boundary of convergence. Since the series converges for $$|x - 3| \u003c 4$$, the radius of convergence is $$R = 4. To $$find the interval of convergence, write the inequality $$|x - 3| \u003c 4 as a $$double inequality: $$-4 \u003c x - 3 \u003c 4. $$Add 3 to all parts of the inequality to isolate $$x$$: $$-4 + 3 \u003c x \u003c 4 + 3$$, which simplifies to $$-1 \u003c x \u003c 7. $$The interval of convergence is therefore $$( -1, 7 ). To $$fully determine the interval, you would check convergence at the endpoints $$x = -1$$ and $$x = 7$$ separately, but based on the problem statement, the series diverges when $$|x - 3| \geq 4$$, so the endpoints are not included.

Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.2.6

Chapter 11, Problem 11.2.6

Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.

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Identify the center of the power series from the given inequality. Here, the series converges when $|x - 3| < 4$, so the center is at $x = 3$.

Recall that the radius of convergence $R$ is the distance from the center to the boundary of convergence. Since the series converges for $|x - 3| < 4$, the radius of convergence is $R = 4$.

To find the interval of convergence, write the inequality $|x - 3| < 4$ as a double inequality: $-4 < x - 3 < 4$.

Add 3 to all parts of the inequality to isolate $x$: $-4 + 3 < x < 4 + 3$, which simplifies to $-1 < x < 7$.

The interval of convergence is therefore $( -1, 7 )$. To fully determine the interval, you would check convergence at the endpoints $x = -1$ and $x = 7$ separately, but based on the problem statement, the series diverges when $|x - 3| \geq 4$, so the endpoints are not included.

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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 is the distance from the center of a power series within which the series converges. It is found by identifying the boundary where the series changes from convergence to divergence, often expressed as |x - c| < R, where c is the center and R is the radius.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It is centered at c and extends R units in both directions, typically written as (c - R, c + R), but endpoints must be tested separately for convergence or divergence.

Power Series Convergence Tests

To determine where a power series converges or diverges, tests like the Ratio Test or Root Test are used. These tests help find the radius of convergence and check endpoint behavior, which is crucial for establishing the exact interval of convergence.

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Textbook Question

{Use of Tech} Approximations with Taylor polynomials

a. Approximate the given quantities using Taylor polynomials with n = 3.

b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.

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