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(a) Use a graph to guess the value of the limit $ \displaystyle \lim_{n \to \infty} \frac {n^5}{n!} $(b) Use a graph of the sequence in part (a) to find the smallest values of $ N $ that correspond to $ \varepsilon = 0.1 $ and $ \varepsilon = 0.001 $ in Definition 2.

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell University

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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.

02:34

Consider $\lim _{x \righta…

So for part A, let's use a graph to find this limit and to the fifth divided by in factorial. So let's go to our graph here. And you can see once you get to about two ten term in the sequence, it looks like they started conversion to zero quickly. So my guess here is that the limit is zero. Yeah, So for part B, we want the smallest value of end such that the following is true in this case. In general, you should write and minus the limit, but from party A we know that this equal zero So we're looking for the smallest value of capital and such that this implies a M wishes already positive is less than epsilon. So for the first case, we'LL do absalon equals zero point one and let's go ahead and find the value. And in the calculator so zero point one, you can see that the first time that happens the first time you're below point one is when n equals ten. However, the definition requires little and bigger than end. So we should really do and equal to nine. And if you take any little and bigger than this And nine, That means you're starting at ten. And that's what we want. Now we'LL do epsilon equals zero point zero zero one So we'LL go ahead and have to zoom in a little more here until we get a point zero zero one showing up. No, zoom in a little more And there we go. Point zero zero one we see it. Now let's go back a little bit to the graph and here we are. We see an eleven, we're still above point zero zero one. But if we go over to twelve, then we're below point zero zero one. So the first time we get below point zero zero one is that twelve? So we should go ahead and make this eleven and then any little and bigger than eleven. This is a problem to say and is bigger than her equals a twelve. This is why we're subtracting what? From the end. Wait. And that results part B

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