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

$ \displaystyle \sum_{n = 1}^{\infty} \frac {n1}{100^n} $

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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

Oregon State University

Baylor University

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.

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Use the Ratio Test to dete…

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you re like tio use the ratio test to determine whether the Siri's conversions air diversions. So this means that we should look at the limit and goes to infinity absolute value and plus one over an where Anne is given by this term here. So let's just go ahead and replace the numerator and denominator. So in this case, we replace and within plus one in our formula here. So that will give you and then in the denominator and which is just this. And I removed that slew value because all these terms were positive, so there's no need for it now. Let's just go ahead and take that blue fraction, flip it upside down and then multiply to the red fraction. So now to make this would be easier. If you look at the one hundred terms here, the corresponding terms weaken right. That is one hundred to the end, and then the denominator. We can write that it is one hundred times one hundred to the end, and we could cancel those off. That will give you just the one over one hundred. And now we have to deal with these factorial terms here. So if you see and plus one fact, Auriol factorial Just go to the definition. This is just a product of all the numbers all the way up to N plus one imagers in the denominator. And so we see, we only cancel the first in terms here, and we're still left with the n plus one. Okay, so when we take that limit will have something approaching infinity divided by hundred. So this will diverge off to infinity. Now, since we're using the ratio test, we see that we have an expression that's strictly bigger than one. And that implies that the Siri's diverges yeah, and urges by the racial cyst.

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