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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.$ 2 + 0.5 + 0.125 + 0.03125 + \cdot \cdot \cdot $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Harvey Mudd College

University of Michigan - Ann Arbor

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.

01:34

Determine whether the geom…

02:09

Determine whether the infi…

01:22

03:04

01:55

00:44

$17-26$ Determine whether …

01:25

Determine whether the seri…

02:17

Infinite Geometric Series …

01:23

determine whether that she had metric. Siri's is converting our diversion, and if it's conversion, find this home. That won't be a problem because we do have a formula for geometric Siri's and here they're giving you the Siri's. So before we look at the serious and detail, let's just go to the side and recall the definition for geometric. This means that you have Ah, some of the form A are to the end. Some people might right they are and minus one. It doesn't matter here whether and also I'm not writing the starting points here, it could be one. It could be zero it actually, it could be any number of I won't even write anything down there. Basically, the idea is each time and increases by one. You're multiplying by the same number each time, and it's always are r. So in this problem they're telling what we know that this is geometric. What we have to find out what our is and what is our A. So here to find the r noticed that if you take this if you take one term and you divided by the previous term, your leftover with are so If you ever want to find our and from a geometric series, just take some term that you see anyone that you like and divided by the one right before. For example, Here I look at the seconds or I'm divided by the one right before it, so I get a half over to which is the fourth. So that's my r and you could check. You could use this for any two values you could use instead of using to one point five. You could use point five and point one two five. If you divide point one two five over playing five, you'LL still get a fourth, so that is our are. And then here we can go ahead and use a to just be to that first term. So here, since our satisfies the following inequality, it's less than one. The Syrians will converge, So this is the condition that you need for convergence. Otherwise diverges even if it's equal to one diverges. Now let's use our formula for the geometric series. So here we have a over one minus R. Or I think the easiest way to memorize this is the first term. If you use this formula here it. This won't depend on what you're starting Point is, And in our problem, that's just two over one minus a fourth. So we have to over three over four, and then you could go ahead and simplify that Teo eight over three. And that's your final answer for this geometric series.

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