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Find the radius of convergence and interval of convergence of the series. $ \sum_{n = 1}^{\infty} n^nx^n $

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interval of convergence $[0,0],$ radius $R=0$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Campbell University

Oregon State University

Harvey Mudd College

University of Nottingham

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.

01:30

Find the radius of converg…

05:58

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01:11

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03:21

03:00

okay for this problem. Our radius of convergence are is lim a as n goes to infinity into the end over in plus one to the end plus one. Okay, so we can rewrite that as limited n goes to infinity of in over in plus one to the power of end and then times one over in plus one. Okay, so in divided by n plus one to the power of n that's one over e and then we're left with limit as in approaches infinity of one over in plus one and this is zero. So we have won over E which is just some finite number time zero So we end up with zero. So the radius of convergence here is zero. The interval of convergence. We just checked the end points on this case. We're just checking zero here and we're checking to see whether or not this is going to converge. There's just adding up zero a bunch of times. So this this is going to be zero to that is going to work. Case will include zero, but nothing else will be included in our interval of convergence Are interval of convergence is just zero all by itself

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