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Use the Root Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 2)^n}{n^n} $

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Absolute Convergence by root test because $L=0<1$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

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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.

03:03

Use the Root Test to deter…

01:43

01:13

02:01

02:48

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00:41

Use the comparison test to…

let's use the root test to determine whether the Siri's conversions of averages Now first, let me go ahead and call this an and know that we can also write a n equals negative to over end to the end power using your laws of exponents. Now, the root tests requires you to look at the end through of a M. So I'll use our one of our formulas here. We know that we could always write this radical as X to the one over. And so here I'LL use this formula simplified for Anne. And then we're raising this to the one over and power and recall that if you're taking an expo nit like we have here eight of the bee and you raise that to another exponents, see, that's just the same as multiplying to be in the sea. So here we'll supply the end and the one over and cancel out your left over with the limit of negative to over end. This goes to zero, which is less than one. So the Siri's converges and that's by the root test. And that's our final answer

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