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Suppose that the radius of convergence of the power series $ \sum c_n x^n $ is $ R. $ What is the radius of convergence of the power series $ \sum c_n x^{2n} ? $

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Radius of convergent is $\sqrt{R}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

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University of Michigan - Ann Arbor

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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.

02:41

Suppose that the radius of…

04:18

01:53

01:57

Suppose the series $ \sum …

06:10

Find the radius of converg…

00:49

What is the radius of conv…

01:54

01:38

Suppose the series $\Sigma…

we can use the ratio tests on this sum here. So lim his n goes to infinity absolute value of and plus one over. And we want this to be less than one by and we mean this whole chunk here, including the X values. This is Linda as n goes to infinity. Absolute value of X to the two times in class one times C n plus one over X to the two n times cnn X to the two and plus one. If we expand that, that's X to the two n plus two. So the X to the two and we'LL cancel out with this X to the two and and we'LL just leave us with X squared And over here we still have the c n plus one overseeing and again we want this to be less than one. We can write this as absolute value of X squared is less than limit as n goes to infinity of absolute value of C n o ver si n plus one. So if we just flip this, multiply both sides by the reciprocal of this limit and since that's a positive number, that's not going to affect this inequality sign here. This limit here should be the radius of convergence. For this sum, as we saw from the previous exercise, That's our Okay, So absolute value of X squared is the same thing as absolute value of X squared. So the absolute value of X squared is less than are Can we get that absolute value of X is less than a square root of are And this tells us that the square root of are is that radius of convergence for this power Siri's

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