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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = 3 - 2ne^{-n} $

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the sequence is bounded

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.

01:50

Determine whether the sequ…

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first, let's show that this sequence is increasing, so the way to do that here is we can define f of X to be three minus two X e to the minus X. Now f is monotone or I should say not F, but an This sequence is monotone if and only if the derivative has the same doesn't change sign, let me work it that way doesn't change signs. So either always negative or always positive, but not both. So here, let's look at F. Take the derivative. So here three goes to zero, then we have minus to either the negative X and then plus two x either the minus X. I use the chain rule there canceled out some negatives, so this is either the minus X, and then we have two plus two x here also because we're dealing with the sequence and it's bigger than or equal to one. So that means we want X to be bigger than or equal to one. That's important because here, if X is bigger than or equal to one, either the minus X is always positive for any X. But then two plus two X is also positive due to this condition up here. So we're multiplying two positive numbers together, so the result is also positive. So this shows that a N is increasing because positive derivative means increasing F increasing f means AM is increasing. So that answers the first part of this question. Now we'll go on to the next part to determine whether or not this thing is bounded. So to do that, I'll need more room. Let me go on to the next page, so limit. So here the question is whether it's bounded the sequence. So the limit as n goes to infinity of the end of term. Mhm. Yes. Now the first part. Well, that's just three. And then let me go ahead and push this inside and let me write that limit as X instead of in. So I could use local tells rule if I have to, and we can see here that if you rewrite the e and without the minus sign, you have infinity over infinity and the limit, so we should use low petals role here. So this is by low Patel of that we have a limit of the type infinity over infinity. That's an indeterminate form so low, Patel's rule says. You take the derivative of the top and the bottom, so that's a tool of top. The denominator ends up staying the same. Absolutely differentiate. And then we have two over infinity. So this is zero. So we just have three minus zero equals three, and we could conclude that the sequence is bounded. So we have and is bounded. And the reason why, since its emergence and convergent sequences are always bounded something. So to summarize, we've shown that a and is bounded and monotone increasing mm, mhm, and that's our final answer.

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