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

$ \sum_{n = 1}^{\infty} \frac {(x - 2)^n}{n^n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 8

Power Series

Sequences

Series

Missouri State University

Harvey Mudd College

University of Nottingham

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.

03:00

Find the radius of converg…

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01:54

to figure out the radius of convergence will first just look at where the Siri's converges So we'LL use the ratio. Tests take limit as n goes to infinity of absolute value of A and plus one over a n where a. And is this whole thing here, including the X values. So that's limit as n goes to infinity of absolute value of X minus two to the end plus one divided by in plus one the power of in plus one divided by a n, which is something as multiplying by the reciprocal. Okay, so x minus two to the impulse one divided by X minus two to the end. That just simplifies tow X minus two. Okay, that's one of the nice thing about having a common base. And here we don't quite have a common exponents, but in plus one of the power of in plus one can be written as in plus one to the N times in plus one. So if we do that, then we can rewrite this thing like this and now limit as n goes to infinity of in over in plus one to the end. This is one over e So now we have limit as n goes to infinity of absolute value of X minus two over in plus one. And this value is one over e. So he is just some number here and one over in plus one goes to infinity. Doesn't matter what exes as long as exit any finite number. This is just going to go to zero, which is certainly less than one. We're going to get convergence regardless of what value of X. We're looking at here. So the radius of convergence is infinity in the interval of convergence is minus infinity to infinity.

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