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Find the radius of convergence and interval of convergence of the series.
$ \sum_{n = 1}^{\infty} 2^nn^2x^n $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 8
Power Series
Sequences
Series
Baylor University
University of Nottingham
Idaho State University
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.
02:45
Find the radius of converg…
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for the radius of convergence. We take the limit as n goes to infinity Absolute value of a n over a n plus one. And we want for this to be less than one. The ends here is this to to the end times and squared. Okay, so up top, we're gonna have to the end and squared down in the denominator We have to do the in plus one times in plus one squared. So to the end about about to the M plus one is one half and then n squared over in plus one square it is, and over in plus one squared limit as n goes to infinity of n over in plus one is one this turns into one squared so one half times one squared would just get one half So one half is the radius of convergence. So this is our are for the interval of convergence. We need to figure out whether or not we include minus one half and whether or not we include one half case, we need to check to see whether or not we get convergence for those values. Okay, so this was the sum that we had we had to the end. I'm in squared times X to the end, and we were going from an equals one to infinity. So one two check are in points. So we needed the check Well, minus one half. So plug in minus one half here and see whether or not we get convergence. So when you plug in minus one half for acts, we get minus one half to the end. So if we rewrite that a little bit, we get minus one to the end times one half to the end. So to the end, one half to the end cancel. And this should be clear that this is not going to converge because the terms do not go to zero. So that is going to diverge. And similarly, when exes one half the terms that we get are going to be two to the end and squared one half to the end. So the terms that were going to be summing up is just in squared, which clearly do not go to zero. So we can't possibly have convergence, So that's also went to diverge. So both of the end points we have to toss out so the interval of convergence. We throw out both minus one half and one half and just leave it as this open interval here.
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