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(a) Determine whether the sequence defined as follows is convergent or divergent:$ a_1 = 1 $ $ a_{n + 1} = 4 - a_n $ for $ n \ge 1 $
(b) What happens if the first term is $ a_1 = 2 $ ?
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
Oregon State University
Harvey Mudd College
Baylor University
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.
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whose problem it helps to just write out the first few terms. So we know a one is one Hey, two is going to be four minus a one. So four minus one, which is three pay three is going to be four minus a too. So four minus three, which is going to be one. And from here, we know that the sequence should just continue in this way. One, three, one. But, you know, if you really want to convince yourself, you keep going. I thought I was going to be safe. Four minus a three is four minus one, which is three. A five is four minus a four four minus three, which is one. So this just switches between one and three, so that's definitely going to diverge. Okay. And once we got Tio Tio here, we knew that it was just going to switch between one and three, because each term is just defined only by the previous term. Okay, once you've gotten to hear, we've already been in this situation once before, right? So we know that if one is the previous term, that next term has to be three. So once we get to this one. We know that the next term has to be three. Okay, And then once we're at three, we know that the next term has to be one, because again, each term is defined only by the previous terms. So if we've been in the situation where we've gotten one before and the next term was three, we know that if one shows up again, the next time has to be three. And likewise, if three shows up in the next term is one, if that ever happens again and the three shows up, the next term is going to be one. But it's always good to, you know, you really convince yourself of it by, you know, right, Not a few more terms and seeing that indeed it is just going to alternate between one and three. So certainly divergence
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