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Suppose the series $ \sum c_n x^n $ has radius of convergence 2 and the series $ \sum d_n x^n $ has radius of convergence 3. What is the radius of convergence of the series $ \sum (c_n + d_n) x^n? $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Campbell University

Harvey Mudd College

University of Nottingham

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.

01:38

Suppose the series $\Sigma…

01:49

Suppose that the radius of…

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04:18

since this sum here has a larger radius of convergence and this sum, we know that these dian terms we're going to be approaching zero significantly faster than these see in terms. So much so that as n goes to infinity, Deanne overseeing is going to go to zero. Now we can use the fact that our radius of convergence is the limit as n goes to infinity of absolute value of C N plus stian over C n plus one plus D in class one. And now we can divide both sides both the top and the bottom by CNN to get one plus D and overseeing over C N plus one oversea end plus the end plus one overseeing as we mentioned, since these dian terms go to zero much faster than these see in terms, this is going to go to zero. This is going to go to zero and then we'LL just have one divided by C n plus one over seeing plus one. So division is the same thing as multiplying by the reciprocal and so we get here. Okay, but this should be the radius of convergence for this sum here, which we set is too. Okay. So in fact, whenever you have a sum, some power Siri's and some other power Siri's and you're adding them up in this way, that radius of convergence for the sum is just gonna be the smaller of the two unless they happen to have the same radius of convergence, In which case, there's not quite as much to say.

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