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Use the Root Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac { - 2n}{n + 1} \right)^{5n} $

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Divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Missouri State University

Campbell University

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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.

01:43

Use the Root Test to deter…

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01:40

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02:22

let's use the root test to determine whether the series converges or averages. So here, let me go ahead and circle this term and call this my a n. The root test requires that we look at the limit and through of a n so recall the following fact. We can always write a radical and through as X to the one over and power. So here's our a n and then we're raising that whole thing to the one over and power mhm. And so we'll use another fact here about exponents. So here in the parentheses, we have and express and exponents a negative two and over n plus one, and that's all being raised to an exponent. So that's our baby. And then that is being raised to another exponent, one over end. So the rule is is you can just multiply those exponents. So, in our case, well, just multiply the one over N in the five end and the ends cancel. And here as well, we should technically be looking at the absolute value of a M. And you can see why this is important here because which will need the absolute value here. So let me go out and just put my absolute values back here. And then Now, well, you have this fifth power to deal with. So if we wanted, we could drop the absolute value. Now, just make sure that you dropped the negative as well. And then two to the fifth. It's 32 and we'll have an unfit there. But we also have an N plus one to the fifth, and you can you can rewrite these ends as an over and plus one to the fifth. And when we take the limit, that just goes to one to the fifth, which is equal to one. So in the limit does go to one, and we just have 32. This number is bigger than one. So by the root test, we conclude that the series diverges, So diversion is our answer.

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