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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 0}^{\infty} \frac {3^{n + 1}}{(-2)^n} $

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The geometric series diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

University of Nottingham

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.

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Determine whether the geom…

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Determine whether the seri…

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it's determine whether or not this geometric series converges and if it's commercial, will go in and find that some as well. So here we can rewrite the Siri's and equal zero to infinity. Let me go ahead and pull out one of the factors of three here, so I'm going to do here is use the fact that three the M plus one, it's just three and times three. So I'm gonna pull on a three there I have my three to the end still and then on the bottom I still have that minus two to the end and then once more I'll use another laud iPhone. It's three and then up. What I'LL do here is all used the fact that eight of them over B to the end is a over B to the end. So using that here with a equals three B equals negative, too. We can write it in this form and now we see that this Siri's look something like an equal zero to infinity a are the end. So this is usually how geometric series are reading the signal notation so we could see that the A is three. And here we see that the R is negative. Three halves, so for geometric series. So let's write the following. We have since the absolute value our which in our case is positive three over too satisfies this and really it could even be equal to one. Any time. Your absolute value are is one or more, which is what we have here. The series will diverge. So let's say the Siri's is diversion. Therefore, we don't have to find the sum so we could ignore the second sentence so divergent and that's my final answer.

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