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Find a power series representation for the function and determine the interval of convergence.$ f(x) = \frac {1}{1 + x} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 9
Representations of Functions as Power Series
Sequences
Series
Missouri State University
Campbell University
Oregon State University
Harvey Mudd 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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Find a power series repres…
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Use the power series$$…
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Use the power series $\fra…
for this problem. What we can see is the form of this function is very similar, Teo. A function that we know, which is half axe. Oh, maybe they see J x Kowtow one Oh one ninety six. So it always knows the power. Siri's of Jacks is Howard Siri's Asik Oto Some nation off go from zero to infinity. Nice. Oh, ax to the talent. So there is in this case we notice as thanks ICO toe jay ninety six h is e kowtow while minus niner six So on the final is equal to one classics Wow! Well, then plastics is equal effects so they know all of them. I hear it. So the power series for affects Asik Oto Jay minus tax Because summation on go from zero to infinity of minus X So that's how you end You know this case What we get is some nation go from zero no to infinity enough minus one to the power own times Axe to the Palin on the next step is to find this interval of convergence to find the eighth is is into off convergence If we are If you are no so interwar of convergence for J Axe is actually and minus one. Why on DH I've actually kowtow J minus X so they can we can know as X is convergence in tow is also you're just flip exact Same. So what you got is also mention on the way. So if you don't know, that's probably a it's fine. You can use ritual test What God is Hey, mate. On go to infinity and plus man Absolutely Area off and placement O a n. So in this case, a m e equal to this Asik oto this serious? So what? We have ways as equal to minus one with the power and prestige Times X to the power and class one o r minus v. So how end times asked, however, so take its absolute value What a kite is minus one times x So is ICO tow absolute value of facts So they know a as a racial test. This one must be smaller than that and also X absolute value is all the smaller one and we get the same radio of concordance and into loft conversions. Lee, I don't
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