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Is it possible to find a power series whose interval of convergence is $ [0, \infty)? $ Explain.?
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 8
Power Series
Sequences
Series
Missouri State University
University of Nottingham
Idaho State 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.
02:11
Is it possible for a power…
01:50
$$\begin{array}{l}{\text {…
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Are there power series tha…
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Interval of Convergence De…
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What is meant by the inter…
03:44
you should always be ableto right. A power Siri's in this form. Okay. Where the sum could go from zero to infinity or wondering Cindy er, whatever. Okay. And then, if you have your power, Siri's in this form, then the interval of convergence is going to look something like this so we could get Ah, we might include the end points. We might not, but at least on this open interval, we will get convergence. So we will get convergence here and then the end points. It depends on the power. Siri's okay and are here. Our is the radius of convergence for the power Siri's. So the radius of convergence is one half of the length of the interval of convergence. So here the length of the interval of convergence is infinity. That means that the radius of convergence in this case is infinity. So since sea is just some finite number C minus, infinity is minus infinity and C plus, infinity is infinity. So in other words, if we got convergence here, then we would need to get convergence here
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