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Find a power series representation for the function and determine the radius of convergence.$ f(x) = \frac {x}{(1 + 4x)^2} $

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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

Oregon State University

Baylor University

Idaho State University

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.

02:51

Find a power series repres…

01:32

03:34

01:33

05:55

05:50

Okay, So find a power source. Reputation function and determine the rapist from murders. That's it Goes eggs over cue. The square of one has warrants. So how do we suppose to spend this? Okay, we can do the technic. So basically, this is next. You one horse comes to do with the one word with us for X. So a status. It is true. Well, this is the reality of what, One hour with the spreads. This is just one of you are not the one who are with us for X squared times for services once more, or one of us for a square. And here we have better. When we were for one horse Oh, he finds it out. So this is shoot. Okay, so then things becomes easier. We can just expand this part that is in from zero to infinity. Mine is looking forward to part in an extra hour. So on, Actually, we can We published part into this this old and it becomes too from zero to very on minus four. Okay, minus one to the power plus one, four two. Health on minus one and exited off plus one. Yes. So this the power Siri's and funded with this convergence. So happen here because we expanded from here and we require money. Sport X is from zero to one is from my neck. You want one? So that wass as is from nephew one fourth to one fourth. That is thing, Regis Convergence. Our Equus went forth.

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