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(a) What is the radius of convergence of a power series?How do you find it?(b) What is the interval of convergence of a power series?How do you find it?

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a) The radius of convergence is the distance between the endpoints of the radius of convergence and its center. It is found by adding the absolute values of both endpoints together and dividing by two.b) The interval of convergence is the domain of $x$ values for which the power series converges. It is found by using the ratio test to determine for what values the series is convergent.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Boston College

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

02:28

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

01:52

$$\begin{array}{l}{\te…

02:23

Describe the radius of con…

that we'LL start with the interval of convergence. The interval of convergence is the largest interval for which your power, Siri's converges and the radius of convergence is just half of the length of that interval. So say that your interval of convergence was something like Minus Are our could be open, Could be closed, could be half open Whatever. Ah, then your radius of convergence would just be R skin that you can find something like this by taking your power. Siri's so it's a look, something like this. If you did the ratio test, then you'd see that you would need for Lim n goes to infinity of absolute value of Ace of N Plus one over Ace of End Times X. That would have to be less than one. Okay, but then this is just going to give you some open interval, and then the case is where you get equal the one the test is inconclusive. So you just need to check the cases where X is equal to one. You know, you just have to check that separately. You can use this to figure out what our is a case of This happens then your R value is limit as n goes to infinity of the absolute value of a n over a n plus one. Okay, And then you know that your interval is gonna look something like minus R T r. Unclear whether or not this should be open or close here. Unclear whether or not this should be open or closed. You just have to check the in points separately. So take minus our plug it into your sum. Figure out whether or not you get convergence or divergence. If you get convergence than you want to include that in your interval of convergence. If you got divergence and you'd leave this part open and similarly, you'd have to check our figure out whether or not that should be included are thrown out.

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