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

$ \displaystyle \sum_{k = 1}^{\infty} ke^{- k} $

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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

Campbell University

Oregon State 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.

02:21

Use the Ratio Test to dete…

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let's use the ratio test to see whether or not the Siri's converges. So the ratio test requires this expression a k equals K E to the minus K. So let's look at the limit. Is king goes to infinity? Absolute value a K plus one over a k So the numerator Let's do this in red, they have K plus one e to the minus K minus one. What the denominator? Okay, we know what that is from up here, such as plug that in and we could drop the absolute values here because all the numbers involved, they're positive except the exploring here. But he too, the exponents still positive. So let's look at this fraction over here. That's just one over easy. And then also we have K plus one over again. However, if we take the limit of this, you could do low pitch house rule, for example, and you will get that this limit is equal to one. So here when we multiply these two numbers together, we're just getting one times one overeat, which is one overeat. That's less than one. So we conclude that the Siri's conversions bye the ratio test, and that's our final answer

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