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Evaluate the indefinite integral as a power series. What is the radius of convergence?$ \int \frac {\tan^{-1} x}{x} dx $

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Power series of$$\int \frac{\arctan (x)}{x} d x=C+\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{(2 n+1)^{2}}$$[ Radius of convergence is 1]

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 9

Representations of Functions as Power Series

Sequences

Series

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

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Evaluate the indefinite in…

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$25-28$ Evaluate the indef…

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Evaluate indefinite integral as a power series and well is there this convergence all right, so first we're going to expand this as a power series equals to 1 x times, arc tangent x d x so actually expand this part, and that is equal to and from 1 To infinity x to the power of 21 over 2 n minus 1 times minus 1 to the power n minus 1, this is yeah exactly and we can simplify this and exchange the order integral and some. So it's going to be minus 1 to the power minus 1 x to the power of 2 and minus 2 over 2 and minus 1 the x. So we can continue to simplify this integral, which going to become, is just going to become a extract, minus 1 to the power 1 or 2 and minus 1, and we integral x to the power of 2 and minus 2 x. So at last it goes to at last the integral is going to be x to the power to 1 over 2 and minus 1 point equals to over 2 minus 1 square and the red this convergence. Actually, we spun from here- and we know that arc tangent x, a serious convergence for arc tangent x- is r equals 1.

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