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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {(n!)^n}{n^{4n}} $

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Divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Missouri State University

University of Michigan - Ann Arbor

University of Nottingham

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.

04:14

Test the series for conver…

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in this problem, we have to test a Siris for the convergence or divergence. But the only tricky thing is is we're not told what test use. We kind of have to use our knowledge about Siri's to determine the test. So let's first review the Siri's that were given were given this Siri's from n equals one to infinity of in factorial raise to the end all over and raised to foreign. So let's apply the test for divergence. And let's just see what we get the test for Divergent says that we can take the limit is then approaches Infinity of Ace event. Well, that would be the limit as an approaches infinity of in factorial over end to the fourth, all raised to the end. Now this is a little bit tricky. We have to be comfortable with factorial, so let's just simplify what we have. We'll get The limit is un approaches infinity of an times n minus one times n minus, two times and minus three all over and to the fourth times and minus four Factorial race to the end and then we could simplify that we get the limit is n approaches infinity of one minus one over N times one minus two over end times one minus three over in, and we'll multiply that by end minus four. Factorial all race to the end. Well, now we're in a position that we can plug in infinity to our limit. When we do that, we'll get one minus zero times one minus zero times, one minus zero times infinity all raised to infinity. Well, that's clearly infinity. So that means that we found that our limit approached infinity. So our Siri's is divergent by the test for divergence. So I hope that this problem helped you understand a little bit more about Siri's and how to choose a test and also how we can go about showing the diversions of a Siri's by using the test for divergence.

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