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Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?$ a_n = n^3 - 3n + 3 $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
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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.
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Determine whether the sequ…
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first, let's determine whether it's increasing or decreasing. So here, what we can do is just replace this with the function f of X By replacing all the ends with ex. Now we have f prime of X equals three X squared minus three, and this is bigger than her equal to zero. If X is bigger than or equal to one, so F is increasing. Therefore, a N is increasing there and which implies that it's monotonic. So that's the first part of this question increasing. And it is monotonic. The next question is, is the sequence bounded? Well, notice that the limit as n goes to infinity of a n equals the limit. X goes to infinity. F of X equals infinity. So this means that a N is not bounded since it's not bounded above, so not bounded. But it is monotone increasing, and that's our final answer
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