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Find a power series representation for $ f, $ and graph $ f $ and several partial sums $ s_n(x) $ on the same screen. What happens as $ n $ increases? $ f(x) = \frac {x^2}{x^2 + 1} $

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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

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.

03:34

Find a power series repres…

02:55

10:21

The problem is find a power series, representation for f and graph f and several partial sums, as in on the same screen, what happens at and increases so first, we have to follow in the equation: 1. Over 1 minus x is equal to 1 plus x, plus x, squared plus x cubed plus tot to 1 x. Absolutely value of x is less than 1, so x is equal to x, squared times 1 over 1 minus negative x square since 1, over 1 minus x is equal to sum, and from 0 to infinity x to x, power then will replace x by negative x Square over half this is equal to x, squared times sum and from 0 to infinity negative x square to the power of which is equal to sum and from 0 to infinity, make 1 to the power of n times x to the power of 2 n plus 2 then here is have the value of x is less than 1 and let's look at it. The graph of the function, so the ritecurve is graph of the function f of x in the blue curve is so green curve is as 4 and this our. In truth, this is a sellonso. We can see that as an increases as x becomes a bit approximation to the function f of x when x is between negative 1 and 1.

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