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Find a power series representation for the function and determine the radius of convergence.$ f(x) = x^2 \tan^{-1} (x^3) $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

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.

05:40

Find a power series repres…

01:33

05:55

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03:34

Okay, So fun. The power Siri's for death backs and determine its readers of convergence. All right, so we can first, if spend these attendants of x Q part. So this is equal to and from zero to infinity and X cube to the power of to most wanna words who minus one times nothing wanted power from minus What? And we know that we knew it from the Taylor Siri's of octane in X. The riches of convergence is just articles one. And so this is gonna be seen with five to and from one suing for the X to the power of six and miners, three plus two is minus one over two months. One times ninety one off on minus one. All right.

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