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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 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

Campbell University

University of Michigan - Ann Arbor

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.

02:02

Determine whether the seri…

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01:11

for this problem. We are going to use the ratio test and first I'm just going to simplify the ratio and then I'll take the limit. So we want to know is what does next term that is the K plus one term look like when it's divided by case turns. So I'm going to distribute on the numerator and split the fraction up so I can simplify things down. So I get Kay need the negative K minus one over K to the negative k. Let's eat the negative K minus one over. Okay, even negative K. Now the left hand side reduces to just mhm the negative one and the right hand side. It becomes each the negative one over K. Now, if we take the limit, this K goes to infinity of this we say that either the negative one over K goes to zero. And so we are left with just the two, the negative one and either the negative one is less than one. So the series is convergent by the ratio test

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