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

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Campbell 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.

03:02

Determine whether the seri…

00:49

02:39

01:48

02:18

03:51

let's determine whether the Siri's converges or diverges. So, first of all, I'LL claim that this is a This sum is less than or equal to the sum from one toe infinity of one over and minus one time's end. Now the reason for this is because, and factorial is equal to one times to all the way up Tio. And so this means that in factorial is larger than and minus one times in and Sense and factorial is larger. But over here we see it's in the denominator so the inequality will go in the other direction. So in other words, this fraction is larger because it's denominator smaller. That's what this show's over here. And then now we can use comparison test. But in order to do so, we should see that this Siri's comm urges. So for this one, there are many ways to go. You could try to use the Lim comparison test, so we're looking at the Siri's that's boxed in red. So let that let this term B an and then let it be in, just be one over and square. Then let's look at the limit of a N over beyond this is the limit comparison test. That's just and swear over and times and minus one. And let's evaluate this. You could use low Patel's rule here if you want. Instead, let me just go ahead and divide top and bottom bye and square. And then I get limit one over one, minus one of her head. Now let's go ahead and take that limit one minus one one over one minute zero, which is just one. And then we know that this Siri's will converge. That tells us that this Siri's one over and minus one times in converges. So this is using the limit comparison test. L see Teo abbreviate that now we can use the direct comparison test to explain why our Siri's circled on blue convergence. So, since first of all, we should point out, as the theory states, that we're dealing with a Siri's with positive terms so sense this one over and Factorial is always positive. We've shown that this Siri's converges by the direct computers and test. So this is not the limit comparison. This is the usual comparison. So we used both comparison test in this problem, but we only use the Lim a comparison to show that the larger Siri's convergence and once we realize that by the direct comparison that tells us that our original series converges and that's your final answer.

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