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Find the radius of convergence and interval of convergence of the series.

$ \sum_{n = 1}^{\infty} \frac {x^n}{n^44^n} $

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$$[-4,4]$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

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.

03:01

Find the radius of converg…

02:40

01:11

07:52

02:12

02:57

03:21

04:40

05:58

Hey, to figure out the radius of convergence, we do limit as n goes to infinity of absolute value of a N over absolute value of and plus ones, absolutely of this whole thing where the ends air just one over into the fourth times for the end. This is limit as n goes to infinity Ah, in plus one to the fourth times four to the end, plus one divided by into the fourth times for it to the end. So this's limit as n goes to infinity of thin plus one over in to the fourth and then for the n plus one divided by four to the end Oops, sorry. I forget that for the M plus one divided by four the end that's going to cancel out just before and then and plus one over in as n goes to infinity and plus one over n goes toe wants This is one to the fourth, which is just one. So one times for so just get for here. So four is the radius of convergence Here for the interval of convergence, we need to check whether or not four works when we plug it in here and whether or not minus four works when we plug it in here. So when axes equal to minus four, then we have minus four to the end over into the fourth times for the end. Sorry for the bad hand writing. Okay, so minus four to the end, divided by four to the end, that's going to turn into a minus one to the end because minus forty and is minus one of the end times for the end and then four to the and will cancel out before the end that we have down there. Okay? And then this will converge, but the alternating signed test. And if we look at, if we look to see what happens when X is equal to four, then we would end up with for the end divided by four to the end. So those would cancel. And we just have one over and to the fourth. Since four is a number that is larger than one, this would be something convergent. So this would also converge. Okay, that works. So minus one works are starting minus four works. When you plug it in for works, when we plug it in. So We include both minus four and foreign R Interval of convergence. So we use these close brackets. Tau show that we are including them and that's gonna be our answer.

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