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Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^{10}}{( - 10)^{n+1}} $

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Absolutely Converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Missouri State University

Harvey Mudd College

Baylor University

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.

03:31

Use the Ratio Test to dete…

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03:44

let's use the ratio test to determine whether the Siri's conversions or divergence to do that, We should look at this term Over here, this's our end. So let's end to the tenth Power Overnegative ten to the end, plus one power. And the ratio test involves this limit of the absolute value and plus one over. And and the answer to this limit will tell us whether the Siri's Khun may tell us whether the series conversions of laboratories now in the numerator Let's do that and read So we'LL have n plus one to the ten negative ten and Plus two still have the slum it over here as and goes to infinity for the denominator. Let's do that in Green and we just use our formula over here. Now let's go ahead and rewrite this before we take the limit we have and plus one to the ten and to the ten. And then here we could drop the absolute value. Just make sure you replace these minus signs, so we'LL have ten to the N plus one over ten to the and plus two. So for this fraction on the right, most fraction you could cancel all the tens up there and then you'LL have one left in the bottom. And also, if you take the limit of this expression over here, you can also write. This is and plus one over into the tent, and a limit of this is one to the ten, which is one. So we have one times one over his head. That's one over ten, which is less than one and therefore the Siri's converges by the ratio test, and that's our final answer.

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