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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{n^n} $

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converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Baylor 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.

03:02

Determine whether the seri…

01:24

02:18

02:07

let's use the comparison test to show that the Siri's converges. So here we have in factorial over into the end. So this is one times two times three all the way up to end than end and all the way up to end and times. So this is one half two thirds one number end two over end and over end. And this is less than or equal to one over end times two over and sense three over end all the way up to an hour. And these are all less than or equal to one so we can replace the sum with two over and swear. And this convergence, this is you could use the pee test with P equals two. And that's David than one, which implies conversions now by the comparison test. Our Siri's also converges

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