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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) = \ln (1 + x^4) $

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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

Harvey Mudd College

Baylor University

Idaho State 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.

03:34

Find a power series repres…

02:34

10:21

The problem is find a power serious representation of your half and the graph on several partial psalms on the same screen. What happens as any increases. So first one over one plus X is equal to one over one madness. Negative attacks, which is you got you some. I'm from zero to infinity. Negative X to the power off in your after a while You fax is fast. What? So out in Juan plus tax You can't you into girl off one over one plus X which is equal to some problems, you know, three vanity next you'LL want to the part ofthe end halves X to the power plus Juan over and plus plonk Last constant number. See, we put X zero in this equation so we can find a C is equal to help and one plus zero which is equal to zero. So we have Ellen one plus X. It's like watching some from zero to infinity Next one to the power ofthe end house X to the power ofthe plus one over and plus one. Then we replace axe exc the power of floor so half hour how in one class X to the power off? Or is this going to sound from zero to infinity? Make you want to the power off in terms Axe to the powerful and last four over and plus one. Yeah, absolute value ofthe ACS is the last one. This is a box. Now let's look at this graph over after rocks. No, we can't not the graph off as one as two, Andi asked. Really the paras arms, as one has to ask three. Notice that as an increases as an ax becomes oblige approximation to the Apple Jacks Well, acts between make two one on one.

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