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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 {6 \cdot 2^{2n - 1}}{3^n} $

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The geometric series diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Zakir U.

May 7, 2022

fimd a.b if a=80, b=50 , the angle between a and b is 3?/4

fimd a.b if a=80, b=50 , the angle between a and b 3?/4

Harvey Mudd College

Baylor University

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.

02:52

Determine whether the geom…

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Determine whether the seri…

06:17

Let's determine whether this geometric Siri's converges or diverges and if it's come urgent will go ahead and find the summer as well. So geometric series is usually ran in the form you have, like some number A here, and then you have some term R Let's constantly being well supplied. And so the Siri's could start at any number of many times. Like in our problem, it's one and it goes up to infinity. So let's just rewrite our problems so that it looks like this. So I have six here and let me rewrite this. We have two of the two groups two and minus one. So this is to the two and over to to the one which is to square to the end over, too. So let's write that here we have two square and over to, and then we're still dividing by three end. Let's simplify six over to that's history and then combining these two terms right here we have for over three to the end. So now this were in this form here, and we see that are equals four or three a equals three. However, we know that a geometric series so geometric. Siri's converges on ly if the absolute value of our is less than one. But in our case, since the absolute value are equals for over three is bigger than one, the Siri's will diverge, so the Siri's is diversion, and that's your final answer.

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