How To Find The Interval And Radius Of Convergence

TL;DR

Calculate the interval and radius of convergence for a given series using the ratio test and check the convergence at the end points.

Transcript

we're being asked to find the interval and radius of convergence so let's do it solution now for this problem I'll assume that you know all these series tests we'll go over them but um you know not not in a ton of detail so to find the uh interval of convergence and the radius we'll use the ratio test so we'll start by taking the limit as n approac... Read More

Key Insights

• 🥳 The ratio test is a useful method for finding the interval and radius of convergence for a series.
• 🥳 Checking the end points is necessary in order to account for cases where the ratio test is inconclusive.
• 🧑‍🏭 The radius of convergence is determined by the absolute value of the factor multiplied with the series.

Questions & Answers

Q: What method is used to find the interval and radius of convergence?

The ratio test is used to find the interval and radius of convergence in this content. It involves taking the limit as n approaches infinity of the absolute value of the ratio between two consecutive terms of the series.

Q: How is the interval of convergence determined?

The interval of convergence is determined by finding the values of x that satisfy the inequality obtained from the ratio test. In this content, the inequality is |x - 3|/4 < 1, which simplifies to -1 < x - 3 < 4. Adding 3 to all sides gives the interval (-1, 7).

Q: Why do we need to check the end points?

Checking the end points is necessary because the ratio test is inconclusive when the limit is equal to 1. By substituting the end points (-1 and 7) into the series, we can determine if the series converges at those points.

Q: How is convergence determined at the end points?

Convergence is determined by analyzing the series at the end points. The series is convergent at -1, resulting in a bracket at that point. However, it diverges at 7, resulting in a parentheses at that point.

Summary & Key Takeaways

• The content explains how to find the interval and radius of convergence using the ratio test.

• The ratio test is used to determine the value at which the series converges.

• The end points of the interval are also checked to ensure convergence.