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.

What is the general form of a power series?

$\sum^{\infty}_{n=0}a_n(x-r)^n$

Write the formula for the Ratio Test.

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

What condition on L in the Ratio Test implies convergence?

$L < 1$

What condition on L in the Ratio Test implies divergence?

$L > 1$

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.

What is the general form of a power series?

$\sum^{\infty}_{n=0}a_n(x-r)^n$

Write the formula for the Ratio Test.

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

What condition on L in the Ratio Test implies convergence?

$L < 1$

What condition on L in the Ratio Test implies divergence?

$L > 1$