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

Harvey Mudd College

Baylor 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.

02:21

Find a power series repres…

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So first we're gonna rewrite this form as x, plus 2 minus 3 over x plus 2, and it turns out to be 1 minus 3 over 2 plus x, and this is just 1 over 1 minus 3 over 2 times 1 plus half x, and we continue Doing this so span, this part becomes n from 0 to infinity minus 1 to the power of n x to the power of n and 2 to the power of n. So we can rewrite this form, as 3 times to 0 is greater than or equal to 0 minus 1 to the power x to the power of and 2 to the power plus 1 point. And finally, the interval of convergence is going to be so negative. Half of x is less than 1 implies. The absolute value of x is less.

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