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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = \frac{1 - n}{2 +n} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
Oregon State University
Harvey Mudd College
Boston College
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.
02:37
Determine whether the sequ…
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let's determine whether this sequence is increasing or decreasing. Or perhaps it's neither. In which case we say it's not monotone. Then we'LL see whether or not it's founded, so to show that it's monotone well, we can do is try to show. For example, if we'LL claim that it's decreasing, which will be the case here, then this means that that the following term is no larger than the previous her for all in. So we can go ahead and show this by algebraic Lee by hand, or we can look at the function. Sometimes this is just more convenient to use. Let's go ahead and use X so we can use some calculus, take derivatives and so on. Then here we're defining f such that F event equals an so to show that the end sequence is decreasing. We'LL just show Epps decreasing and to show ifs decreasing well to show that the derivative is negative. Okay, so let's go ahead and compute that derivative f prime of X. We're using the quotient rule here, So you recall the question rule don't. So we go ahead and take the derivative of the numerator so it's minus one and then times the denominator minus the numerator times, the derivative of the denominator over the denominator squared. Now that numerator loose that's minus two, minus X minus one plus x So those exes will cancel and I have negative three over a square, and this is always negative, that denominators positive. But the numerator is negative, so the fraction is always negative. So this allows us to conclude that the sequence and is increasing. So now the next part is whether or not this thing is actually bounded. Let's go to the next page. Since a M is decreasing, we have at the limit of, and if it exists, it's less than or equals who a N is less than or equal to a one. So this is due to the fact on ly that it's decreasing a ones the largest. You're only getting smaller after a one. This is why I am is less than or equal, say one. And since the sequence is continuing to decrease, the limit, if it exists, will be the smallest of them all. And for this problem we have a one that's equal to zero. We can go ahead and find a limited a and just buy using some algebra or Lopez House rule. In either case, this is negative one so that we've just shown that A N is between negative one and zero. This is for all in. Therefore, the sequences also abounded. So going on to the next page to summarize and is bounded and is decreasing, and that's our final answer.
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