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If $ \sum a_n $ and $ \sum b_n $ are both divergent, is $ \sum \left(a_n + b_n \right) $ necessarily divergent?

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Oregon State University

Harvey Mudd College

Baylor 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.

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if the sum of a n and the sum of being or both diversion is the sum of a M plus being necessarily diversion? Well, not necessarily. So here. Not necessarily. Sometimes it is. Sometimes it converges. Not necessarily. So hear my example. Let's take a N to be two and let's take being to be minus two. Then we have a n equals to this diverges No, by the divergence test. So if you take here the limit of a N, it's just a limit of two, which equals two, and it's not equals zero and anytime the limit of and is non zero, that means it diverges. Similarly, this bee in Siri's here also diverges by the diversions test. So we have two diversion Siri's and in B end. However, what happens when we look at the sum of a and plus being well, if you just write this out am is too bn is negative too. So we just get an infinite sum of zeros and it doesn't matter how many zeros we add. Since all of these numbers are zero, the entire sum will be zero and that before it converges, so it is possible for the sum of and to be and to be convergent so that it's not necessarily diversion, and that's our final answer.

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