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(a) What is a convergent sequence? Give two examples.(b) What is a divergent sequence? Give two examples.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

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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.

01:20

What is a convergent sequ…

03:13

(a) What is the difference…

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$\begin{array}{l}{\text { …

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\begin{equation}\begin…

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Give an example of diverge…

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Give an example of a bound…

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Give an example of a diver…

a convergence sequence is this sequence with the terms a n have the property. That limit is n goes to infinity of Anne. This limit exists and is finite. So two examples of that would be a and e equals went over in where our sequence starts At n equals one. So the first few terms would be one one half of one third You got that out. So this is definitely going to be a convergent sequence. This limit is zero. So the limit exists and zero is definitely a finite number. Another example would be and it is an over in plus one. So Lim is n goes to infinity of a N in this case is just gonna be one and again one is something that's finance and this limit exists. So that is also going to give us Ah convergent sequence. Divergent sequence is any sequence that is not convergent. Okay, so, for example, the sequence to find by and equals And so the first few terms or one, two, three I can't get that here we have limited n goes to infinity of a N is equal to infinity, So infinity is not finite so this not be a convergence sequence. Therefore, by definition it is. It is a divergent sequence. Another example would be sequence to find by an equals minus one to the end of the first few terms would be minus one one, minus one one. In here limit as in goes to infinity of a n. This limit doesn't even exist. The limit doesn't exist, and it also cannot be a convergence sequence and therefore, by definition, is a divergent sequence. So a divergent sequence the terms don't need toe blow up to infinity. If the terms do blow up to infinity than it will be a divergent sequence. But you can also have divergent sequences that are bounded like this one, where the limit simply does not exist.

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