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Use the Ratio Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {n!}{n^n} $

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By the ratio test, the series is absolutely convergent.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

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Use the Ratio Test to dete…

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let's use the ratio test to determine whether the Siri's conversions. So here, let's look at this term and a n equals and factorial over end to the in power. Now the ratio test. I suggest that we look at this limit here absolute value and plus one over. And so let's do the numerator and red So that will be and plus one factorial over end, plus one and plus one. Then that's being divided by AM A N is and factorial over and stand. That's right, the absolute value. You could ignore the absolute value because all the other fractions here consist of positive numbers. So now let's go ahead and simplify this. So let me rewrite this. And plus one factorial is and factorial times and plus one. Then I also have this and to thee and power Then I'LL have this term here. So let me write this as and plus one to the end times and plus one and we still have one term here and factorial. Now it's good and cancel on as much as we can. We see those factorial is go away and plus one terms cancel So we have the limit and goes to infinity and to the end, over and plus one to the end. Now I'm running out of room here. Let me take this to the next page. Now that's free, right, This's end over and plus one to the end. Now, this looks familiar. If you consider this fraction over here, the reciprocal split up that fraction and was even put it to the end power. If you recall this expression here, if you take the limit, this is equal to the number E. So this is the reciprocal of our terms. So that just means that our limit just goes toe one over here. Now this number's less than one. So we can conclude that the original Siri's the sum from one to infinity and factorial over into, then conversions by the ratio test. And that's your final answer.

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