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Show that if $ \lim_{n \to \infty} \sqrt[n]{\mid c_n \mid} = c, $ where $ c \not= 0, $ then the radius of convergence of the power series $ \sum c_n x^n $ is $ R = 1/c. $

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$$R=1 / c$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Campbell University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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.

01:36

Show that if $\lim _{n \ri…

01:35

03:19

Prove that if the power se…

03:22

Suppose that the power ser…

02:01

$$\begin{array}{l}{\text {…

this problem is basically just applying the root test. So this is the sum that we're looking at here is going to do in the root test. We take the in through of this chunk here rather the in threat of the absolute value of all that stuff. So the in through of the absolute value of X to the end, it's just going to be the absolute value of X. I can and the stuff we have leftover is thean through of absolute value of CNN. But we're given that the as in goes to infinity the in through it of the absolute value of C N is equal to C. Okay, so now we have absolute value of X time, see, and similar to the ratio test, we want this to be less than one and since he is non zero, we know that we're allowed to divide by sea. Now we have absolute value of X is less than one oversee and since he is positive since the end through it of ah, positive number is always going to be positive, we know that dividing by sea is not gonna affect this inequality sign. If we're dividing and multiplying by a negative number than this inequality sign would have to be flipped since he is positive. This is less then and this is still less than Okay, So this tells us that we get convergence when absolute value of X is less than one Oversee. So exes trapped between minus one, oversee and see. So we get convergence. A cz long his ex is somewhere between here so we can see that the length of our interval of convergence is one. Oversee minus minus one. Oversee, so to oversee. So the radius of convergence is half of that. So radius of convergence is one oversee, which you could probably just see from looking at this and then the only other place we could possibly get Convergence is if we have equal toe one here so similar to the ratio test. If it's equal the ones and the test is not conclusive, you could get convergence and you might not get convergence. But whether or not we include the end points minus one oversee and one oversee for interval of convergence, it's not going to affect the length of that interval. So regardless of whether or not we include the end points, the radius of convergence is going to stay at one oversee

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