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Determine whether the series is convergent or divergent.$ \displaystyle \sum_{k = 1}^{\infty} ke^{-k^2} $

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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

Oregon State University

University of Michigan - Ann Arbor

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.

01:36

Determine whether the seri…

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00:48

for this Siri's. We shall use the comparison test So the Siri's we can rewrite as from K equals one infinity of okay e to the negative k to the K. And what's important is that either the negative k to the K you know, really would be on the denominator. Sophie, write it that way. It might make it clear what we're about to do, since either the K to the K is always going to be bigger than just each of the k. No turning that off to the side either. The K the K is greater than either the k mom since Kei is greater than one or equal to. So that means that we end up with something smaller than Siri's of just K over you do the K or rewriting it back. Okay, e to the negative k power and this Siri's we know converges Bye ratio test. So if we have they an lesson or equal to be in and being converges, and in this case being you're there are Anne was our original Siri's, we see that are the serious that we want is always smaller than another conversion. Siri's So we have conversion. Bye compares

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