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Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.$ a_n = 1 + (- \frac {1}{2})^n $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
Missouri State University
Campbell University
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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the first few terms that we have here one half when we just plug in in equals one five over for when we're plugging in equals two and bunch of other terms. But it's more helpful. Tohave it written in decimal form zero point five one point two five Your point eight seven five one point o six two five zero point nine six eight Hey, one point o one five six zero point nine nine two two one point. Oh oh three nine your point nine nine eight Oh, one plant Oh, oh, no. Nine. Okay. And then Graff, these we label one through ten down here and then of our values get above one point two five. So we'LL just stop up here at one point two five. So for one, we have zero point five for two. We have one point two five for three. We have zero point eight seven five for four. We have one point o six two five for five for n equals five, we have zero point nine six eight eight for what sees we're on in equal six Now for an equal six, we have one point o one five six okay. And you, Khun, you can start to see that we're reaching some horizontal Assam towed here. All right. These dots are going to be getting closer and closer to this horizontal ask himto so it it does appear that we converge. Okay, it looks like we're converging toe one. And if we write the limit as n goes to infinity of a n, we see that, Yeah, we do end up getting one. Because as n goes to infinity, minus one half to the end is going to go to zero because one half is less than one an absolute value. So if you keep multiplying it by itself, then it's going to approach zero.
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