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If $ f^{(n)} (0) = (n + 1)! $ for $ n = 0, 1, 2, . . . , $ find the Maclaurin series for $ f $ and its radius of convergence.

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convergence $R=1$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 10

Taylor and Maclaurin Series

Sequences

Series

Missouri State University

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.

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Okay, so we're asked to fund medical serious for half and its release of convergence. So what is Thie McLaren's serious flap necks at axe? Because two zero at his fines. So this is just the constrictive at Max as euro over in pictorial and X minus zero to the power of end and of horses from zero to infinity. Okay, so which part in this for murder? And we shall fund that. This is on President Victoria over in Victoria. So this is Gilles. It gives us an plus one times X to power if in. And what is this? So this is a derivative of that's the power of plus one, the extra power from this one. The X and equals with Justo Extract out. This the fish is operator. So this is extra power plus one and his friend zeros for infinity and equals to the derivative of Earth. Next comes this part's actually power of n and A from zero to infinity. And this is our familiar. So it was two x or one minus x o. So our final answer is just to find the virility of this part. And this is just equal to woman's as to the power too womans at minus once, one times x. So it becomes too. One hour square of woman is X and okay fund the readers of Convergence. The rarest convergence is going to be the absolute events. The absolute bell of axe is less than one, so that is the greediest of convergence.

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