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Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $$ \displaystyle \sum_{n = 0}^{\infty} e ^{nx} $

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$$\frac{1}{1-e^{x}}$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

Harvey Mudd College

University of Nottingham

Boston College

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.

02:36

Find the values of $x$ for…

03:17

Find the values of $ x $ f…

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01:20

Find all values of $x$ for…

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02:05

02:37

Let's find the values of extra wish, the Siri's comm urges. So that will be one part of this problem here. Let's call that part one. And then for those values of X in part one. Well, go ahead and find the sum So that'LL be the second part of this problem. So for the first part here, let's just rewrite the Siri's. It's actually geometric, but it's it might not be obvious by the way, that it's Rin. So here I would just right This is either the eggs to the end power here. Understood. Using your laws of exponents you could always rewrite. This is a B times C. So if you want, you could always pull out one of the exponents here and I just chose to pull up the end. So now I can see that this is geometric and R r is equal Teo either the ex So once excess fixed, this is your r value and we know that geometric that these will converge, converge if we have the absolute value are is less than one. So in this case, this will converge. If the absolute value of either the ex was just either the access is always positive, so the absolute value is always equal to itself. And we need this to be less than one. So now we have an inequality here that we need to solve for X because in part one, we do want the ex values. So let's take the natural log of both sides here. So this's since the log function is increasing, the bigger the importance than the bigger the log will be. So since one is bigger than you, fax Ellen of one will have to be bigger than Ellen either. The ex. Yeah, so here on the left side, using the fact that Ellen and E. T. The extra inverse is this is just X and Ellen of one zero or so That's our answered apart one. We want X to be a negative number, and then the series will converge. Now we go to Part two, where we actually find the sum of the Siri's. Assuming that, of course, except a negative number. So it's gonna part two. Now let's hasten room here. So for part two for geometric series, we know that there's some equals. You take the first term of the series over one minus the are the common ratio. So in this problem, the first term happens when you plug in and equal zero. They're so plugged that in, friend and you get a one, you get E X. But that's all being raised with zero power. So you just get one and then one minus R, which is just east of the ex. That's from our are Step One, and that's our final answer. And of course, the sum only is true when we're assuming that exes less than zero, because that's when the series converges, So that's our final answer.

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