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The graph of $ f $ is shown.(a) Explain why the series $ 1.6 - 0.8(x - 1) + 0.4(x - 1)^2 - 0.1(x - 1)^3 + \cdot \cdot \cdot $is not the Taylor series of $ f $ centered at 1.(b) Explain why the series $ 2.8 + 0.5(x - 2) + 1.5(x - 2)^2 - 0.1(x - 2)^3 + \cdot \cdot \cdot $ is not the Taylor series of $ f $ centered at 2.
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Calculus 2 / BC
Infinite Sequences and Series
Taylor and Maclaurin Series
Harvey Mudd College
Idaho State University
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.
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.
The graph of $f$ is shown.…
The graph of IS shownE…
Use the definition of Tayl…
So for this problem, what we have is the Taylor series of F centered one. And what that's gonna look like is ffx equals the sum from an eagle zero to infinity of F to the 10th prime times one over and factorial times X minus one to the end. So that's gonna look like is F one plus F prime of one times X minus one um, plus F prime of 2/2 factorial and then times X minus one squared. So it's going to keep going on. If the series was a Taylor series of F than what we'd have is f prime of one equals a negative 08 However, since the function is increasing for X equals one, this is incorrect. So we know that the given series cannot be a Taylor series of F and then in and to be we want to show it centered it, too, so that this is going to look like is F of two plus F prime of two times X minus two plus F double prime of 2/2 times X minus two squared plus, and that's going to keep going on. But if this series was the Taylor series of F than what we'd end up having is that F F two equals 0.5. But that means that the function is increasing at X equals two. However, that's incorrect because the function actually reaches is it's extremely at X equals two, which would make a prime of two equal to zero, not 0.5. So that means that the series cannot be the proper Taylor series of F.
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