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Numerade Educator

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Problem 2 Medium Difficulty

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

Answer

(a) A convergent sequence is a sequence $\left\{a_{n}\right\}$ whose limit is finite, i.e.. $\lim _{n \rightarrow \infty} a_{n}=$
$L$ where $L$ is a finite number.
(b) A divergent sequence is a sequence $\left\{a_{n}\right\}$ whose limit either is infinite or does
not exist, i.e., $\lim _{n \rightarrow \infty} a_{n}$ is $\infty,-\infty,$ or DNE.

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Video Transcript

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.