Use the Ratio Test to determine the radius of convergence R of ∑n=0^∞ (x^n)/(2^n) . Does it conv

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Use the Ratio Test to determine the radius of convergence R...

Key Concepts

Endpoint Analysis

Endpoint analysis involves testing the convergence of a power series specifically at the endpoints of its interval of convergence. Although a power series converges absolutely inside its radius of convergence, its behavior at the endpoints must be checked individually, as convergence can differ from the interior points.

Ratio Test

The Ratio Test is a method for determining the convergence of an infinite series by examining the limit of the absolute value of the ratio of consecutive terms. If this limit is less than 1, the series converges absolutely; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.

Radius of Convergence

The radius of convergence is a parameter for a power series that defines the interval within which the series converges. It is calculated using methods like the Ratio Test and indicates the distance from the center of the series to the boundary beyond which the series no longer converges.

Divergence Test

The Divergence Test (or nth-term test) is used to determine if a series diverges by examining the limit of its terms. If the limit of the terms is non-zero, the series diverges. This is especially useful at the endpoints of a power series, where terms may fail to approach zero, indicating divergence.

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