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Determine whether the sequence converges or diverges. If it converges, find the limit.$ \{ n^2e^{-n}\} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

Idaho State University

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.

02:38

Determine whether the sequ…

01:52

00:33

in terms that we're looking at is in squared times e to the minus, and it's another way of writing. This is in squared over each of the end, so we should know already that this is going to go to zero because we have polynomial growth in the numerator and we have exponential growth in the denominator. You'd also prove it using low Patel's rule. All right, this will be an easy thing to work with, with for a little bit tells rule. So we're allowed to use low Patel's room when we have something that would give us infinity over infinity or zero over zero if we were to just plug in the value of in here. So we were just to plug in and equals infinity. We would get infinity over infinity here, right? That's something that we're not actually allowed to do because infinity over infinity is indeterminate form. But if we are in that type of situation, we're allowed to use Low Patel's rule, which means that we take the derivative of the top part and we divide by the derivative of the denominator. So we'd get to in, That's the derivative have been squared divided by either the end. That's the derivative of Indian with respect to end. And then we're in the same set up. So we do low Patel's rule again. We get limit as n goes to infinity of two derivative of two and with respect to end is just too. And then the denominator we still hav e the n So now I would have to over infinity, which is zero. So we do converge We converged to zero right And we should know this immediately from looking at it because again, exponential growth is much faster than polynomial growth. So in the end, this either the end is goingto completely dominate over the n squared and the whole day in term is going to end up going to zero.

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