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Define a power series.

A series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$ , where n is a non-negative integer, $a_n$ is a sequence of real numbers, and r is a real number.

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Define a power series.

A series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$ , where n is a non-negative integer, $a_n$ is a sequence of real numbers, and r is a real number.

What is the radius of convergence?

The radius of convergence defines the interval around the center of a power series where the series converges.

Define the interval of convergence.

The interval of convergence is the set of all x-values for which a power series converges. It may include or exclude the endpoints.

What does 'centered at x = r' mean for a power series?

It means the power series is expressed in terms of (x - r), where 'r' is the center of the interval of convergence.

What does the Ratio Test tell you about the convergence of a series?

For a series $\sum a_n$ , let $L=|limlimits_{n→ \infty}\frac{a_{n+1}}{a_n}|$ . If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test is indeterminate.

Explain the purpose of the Ratio Test in determining convergence of a power series.

The Ratio Test helps determine the radius of convergence by finding the values of x for which the limit L < 1.

Why is it necessary to test the endpoints of the interval obtained from the Ratio Test?

The Ratio Test only provides an open interval. Endpoints must be tested individually to determine if they are included in the interval of convergence.

What does the radius of convergence tell you?

The radius of convergence tells you how far from the center the power series converges.

How do power series relate to functions?

Power series can represent functions over an appropriate interval. They are approximations of functions.

What happens when L = 1 in the Ratio Test?

The Ratio Test is inconclusive. Another test must be used to determine convergence or divergence.

All Flashcards

Define a power series.

A series of the form $\sum^{\infty}_{n=0}a_n(x-r)^n$ , where n is a non-negative integer, $a_n$ is a sequence of real numbers, and r is a real number.

What is the radius of convergence?

The radius of convergence defines the interval around the center of a power series where the series converges.

Define the interval of convergence.

The interval of convergence is the set of all x-values for which a power series converges. It may include or exclude the endpoints.

What does 'centered at x = r' mean for a power series?

It means the power series is expressed in terms of (x - r), where 'r' is the center of the interval of convergence.

What does the Ratio Test tell you about the convergence of a series?

For a series $\sum a_n$ , let $L=|limlimits_{n→ \infty}\frac{a_{n+1}}{a_n}|$ . If L < 1, the series converges. If L > 1, the series diverges. If L = 1, the test is indeterminate.

Explain the purpose of the Ratio Test in determining convergence of a power series.

The Ratio Test helps determine the radius of convergence by finding the values of x for which the limit L < 1.

Why is it necessary to test the endpoints of the interval obtained from the Ratio Test?

The Ratio Test only provides an open interval. Endpoints must be tested individually to determine if they are included in the interval of convergence.

What does the radius of convergence tell you?

The radius of convergence tells you how far from the center the power series converges.

How do power series relate to functions?

Power series can represent functions over an appropriate interval. They are approximations of functions.

What happens when L = 1 in the Ratio Test?

The Ratio Test is inconclusive. Another test must be used to determine convergence or divergence.