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If $ \sum a_n $ is convergent and $ \sum b_n $ is divergent, show that the series $ \sum \left( a_n + b_n \right) $ is divergent. [Hint: Argue by contradiction.]

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$\sum b_{n}$ is given to be divergent.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

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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.

01:43

If $\sum a_{n}$ is converg…

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If $\Sigma a_{n}$ is conve…

06:59

Suppose the series $ \sum …

06:20

(a) Suppose that $ \sum a_…

02:31

Given two infinite series …

were given that this sum of am convergence both the sum of being diverges and we'LL use this to show that the Siri's of A and plus beyond his diversion and the hit is that we should argue by contradiction. So let's follow the hidden here. So let's go ahead and suppose that this sum of a M plus B end conversions then if we take this summon, subtract this sum. This will also converge because we're just subtracting two real numbers here. If this some conversions, that means that the sum of a and lesbians a real number. Similarly, we're already assuming that this son commercials that means the sums of real number. So when I subtract, this is also a real number. So here, let me not say conversions. I'll just say this is a real number. However, this Siri's here. We can also rewrite this is and plus being minus and and then here because we're just dealing with real numbers on the inside. Well, you just go ahead and cancel those ends. We have the sum of bien. So on one hand, this expression here, the difference of the two sums is a real number on the other hand, this sum is also equal to the sum of the end. But we're told that this diverges that was the assumptions. So it is impossible for the sum of the end to be a real number and diverge. It's one or the other. If it's equal to a real number, that would mean converges. Excuse me. Therefore, we've arrived at the contradiction, and that's our final answer.

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