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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^{2n}}{6^{n - 1}} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Oregon State University

Idaho State University

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.

02:13

Determine whether the geom…

01:14

02:32

02:11

02:38

Determine whether the seri…

03:45

02:36

03:34

03:23

01:31

Let's determine whether is a geometric Siri's below conversions of beverages and then if it's converges if it converges, we'LL go and find the sun. So first geometric series or easiest to describe when the written in the following form. Because then we know that this thing will converge if absolute value are is less than one. Otherwise it'LL diverge. If absolutely all you are is bigger than your equal to one. So let's find our but to find are we need to get it in this form over here salutary rightists. So I see that the numerator as either of the two and up there apps which I could also actually rewrite that Let me rewrite This is East Square to the end. What I'm trying to do here is get the numerator to the end Power I just did that and also on and the power and the denominator. So what I'LL do here is all right this No. And then I'll just multiply six and then divide by six and that's just six to the end over six. And then I could go ahead and write that as the six down here will come up into the numerator. So six times he swear to the end over six to the end. Now, because I have both of these terms to the end power, I could just pull off that and and write the fraction first and then I have the end power. So now we see that a equal six our equals e squared over six. So we have to approximate with his equals two to determine whether or not this is Weston one or bigger than one. So e to do this, maybe without a calculator. If you have a rough idea of what years he's about two point seven. So here, if we got in square E, you're getting a fell seven point three more or less. And then there we see that this is bigger than six. So this implies east weird over six is better than one. So we have our equals eastward over six bigger than one. So the siri's diverges. And again we're just using this fact here, which I'll circle on blue for geometric series. So diverges and that's your final answer

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