Question

Examples of power series which are convergent ?
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Answer 1

Answer:

Here are some important facts about the convergence of a power series. A power series converges absolutely in a symmetric interval about its expansion point, and diverges outside that symmetric interval. The distance from the expansion point to an endpoint is called the radius of convergence.

Step-by-step explanation:

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Answer 2

Answer:

Determine the radius and interval of convergence of the infinite series

Complete Solution

Step 1: Apply Ratio Test

The ratio test gives us:

The ratio test tells us that the power series converges only when

or

. Therefore, the radius of convergence is 4.

Step 2: Test End Points of Interval to Find Interval of Convergence

The inequality

can be written as -7 < x < 1. By the ratio test, we know that the series converges on this interval, but we don't know what happens at the points x = -7 and x = 1.

At x = -7, we have the infinite series

This series diverges by the test for divergence.

At x = 1, we have the infinite series

This series also diverges by the test for divergence.

Therefore, the interval of convergence is -7 < x < 1.

Step-by-step explanation:

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