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Evaluate the indefinite integral as a power series. What is the radius of convergence?$ \int x^2 \ln (1 + x) dx $
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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
University of Nottingham
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.
03:24
Evaluate the indefinite in…
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$25-28$ Evaluate the indef…
Okay, so you batter the infinite, Integral as a power. Siri's and fun is realism. Convergence. All right, so we can first expend lawn if one plus x. So this is just X minus thanks. Square over two plus x cuba or three minus. That's the fourth over four. And so alone. So the ex assistant X square tons. That's over N to the power. In terms of one on one, it's one trip out on minus one, and it is from one to infinity and this whole thing, the eggs and we change the world and the sub so it becomes and from zero to and from one divinity and x to the power and plus two or in terms that you want to go over. A nice one, the ex and yeah, the universe inside the summation. All right. And what is Thie? And they were off this parts we can pull out now if you want to power. And one is one that we're in sure is from one to even any. And we single you Ballard any roof next to help us to the X. This is just over in times. So thanks to you, power and industry over in three. Yes, so this is our final answer. Forward the power serious expand like you want times and that you want to power and mothers one times extra ball from this three over in times in three and fun. It's, um, readers of convergence. Okay, so where the where the way spend it so expanded from here. Yes, so we could remember the readers of Converters of London because actually, Ari Koh's won and it just applies for this new series. It's called Max. So the readers of Half of X is just Bari who's won.
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