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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ 3 - 4 + \frac {16}{3} - {64}{9} + \cdot \cdot \cdot $

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DIVERGENT

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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.

01:09

Determine whether the geom…

03:18

01:22

05:36

4 Determine whether the ge…

01:40

Determine whether the seri…

00:34

$17-26$ Determine whether …

00:44

let's determine whether this to geometric Syriza's conversion or not. And then, if it's conversion, will go ahead and find the sum. So we know that a geometric Siri's, which is usually written in this form, doesn't have to start at one, but it usually does. Then this will converge on ly if that flu value bar is strictly less than one and it'LL divers, otherwise so absolute value are bigger than or equal to one. So let's find what the R is here. And once we find our will, just rely on this fact here to give us our answer. And if it happens to emerge, well, go ahead and use the formula for the geometric series to find the sun sonar problem. How do we find our well in general? Here's howto find R. If you just go ahead and take any two consecutive terms in the sequence of plugging some end than N plus one and then if you go ahead and divide the second term over the first term over the previous term, everything will cancel out except our so if you ever want to find our and the sum is not given, signal notation just take any element you want except the first one, Let's say, for example, minus four and then just divided by the one right before in our case, dashes three you didn't have to use negative for you could have used sixteen over three, and then you would divide that by negative four. In either case, you will still get minus for over three. So this is our the It's also called the Common Ratio, and you could see why it's called common ratio because dis fact over here. So by this fact, over here we see that the absolute value of our absolute value of negative for over three, that's for over three. That's bigger than one, so it will diverge. So I guess, in summary, our answer is diversions.

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