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Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {x^n}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot\cdot \cdot (2n - 1)} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Idaho State University

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.

06:34

Find the radius of converg…

01:39

case of the radius of convergence is going to be limit as n goes to infinity of absolute value of a N over and plus one where a Is this chunk here without the X value And yeah, maybe we'll well, we'll do it using the ratio test just in case this is confusing to anyone. So the ratio test would do the same type of thing Except we have the sub script with the implicit one of top now and buy this being now we do mean this whole chunk here, including the X values. Hey, so this is going to be X to the n plus one over one times three times that that that times two in minus one times two in plus one minus one. Okay, And then so that's just are being plus one. And we're dividing by being somewhat planned by the reciprocal. So now I have an X to the end over there. Enough top. We're gonna have one times three times that, that that times two and minus one. And this whole chunk here is going to cancel out with this whole chunk here and actually and plus one divided by X to the n is just going to leave us with X. So now we have limits as n goes to infinity of absolute value of X and nothing got rid of this two times in plus one minus one. So we still have that there and for using the ratio test to figure out where we get convergence. We want for this to be something less than one. But notice that this term is going to go to zero as n goes to infinity. So it doesn't matter what value we plug in for X. We're always going to end up getting zero here. Zero is certainly less than one. So X is allowed to be anything for the radius of convergence is infinity, Which is exactly what we would have gotten if we I just did this here. This will short cup would have gotten that R is equal to limit as n goes to infinity of two times in plus one minus one and that would be infinity. Okay, So if the radius of convergence is infinite than the interval of convergences, certainly going to be infinite as well

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