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Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 9)^n}{n10^{n+1}} $

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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

Oregon State 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.

02:42

Use any test to determine …

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01:11

Determine whether the seri…

let's use the ratio test to show that the Siri's converges So we'll be looking at the limit and goes to infinity a n plus one over a end where the end is given by this term over here. So let's first deal with the numerator and plus one. So here will have you noticed that because we're taking absolute values, we can go ahead and ignore the negative nine over here and then we'LL have ten. And then there you were place and with n plus one and then add the other one over here to give you a plus two and then we'll divide that by am absolute value of that. So then, as usual here we'LL take that denominator and blue and flip it over Multiply it And then now we should cancel out as much as we can. We could take off end of these nines and we're left over with the nine when we do this on top. Similarly, we could cancel off and plus one of the tents that will leave you with one on the bottom and then we'll still have our and and plus one left over but in the limit and over and plus one goes toe one. So we get nine over ten, which is less than one. So we conclude that the Siri's the given Siri's converges. Absolutely. And then here we already mentioned what tests the ratio says, so that's our answer.

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