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Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $$ \displaystyle \sum_{n = 0}^{\infty} \frac {\sin^n x}{3^n} $

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Interval of convergence is $(-\infty, \infty)$The sum of the series is $\frac{3}{3-\sin x}$

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.

03:29

Find the values of $ x $ f…

02:06

Find the values of $x$ for…

01:58

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00:59

Find all values of $x$ for…

00:37

Let's go ahead and find the values of X, for which the Siri's converges That's one park. And then for those values of X for which it converges in part one will go out and actually find the summers. Well, that'LL be part two for per one. This is actually a geometric Siri's here, and it's a little easier to see that if you just first write the fraction and then write the exponents outside. Oops, that's not a three out there. That should be an end. And the reason you're allowed to do this is just coming from your laws of exponents a n over being. So here I am just using a equal sign and then be equals three. So if I write it this way, it's easy received at its geometric and that are equals sine X over three. And I know that a geometric series will converge if the absolute value are is less than one. So in this case we need we have absolute value, are which is the absolute value of sign over three. Let's simplify that before we said it less than one. Now let's set this less than one and try to solve this for X. So let's multiply that three to the other side. And this inequality here is always true because we know that always true since we know that sign is always between negative one and one. So if you're always between negative one and one than an absolute value, you're less than or equal to one. And therefore you have to be less than three. So this inequality is true. Oh, for all ex. So here, if we want, we can say negative infinity less than X less than infinity. That's the answer to the first part of the question. All real numbers, all real numbers. That's another way to say the same thing here. That's the interval notation. The inequality notation Negative infinity less than X minus infinity. That's the same thing is saying just all real numbers now for part two. We actually want to go out and find this sum for any value of X that you plug in. We want to find the corresponding some. So we're evaluating this after you pick X so too. Since this is geometric, I know that the sun will always equal the first term and then I divide by one minus are the common ratio. So in this problem, the first term when you plug in and equal zero here you have signed over three. But to the zero power that's just going to give you a one back. And then you have one minus R, which is signed over three. And this thing over here, if you want, you could go ahead and rewrite this as three over three minus sign X. So the way to do this is just tow, get a common denominator down here and then just simplify things. In either case, the Siri's converges for all roll numbers X and for any role, our next that you use this down here in the bottom left is the value of the sum, and that's your final answer.

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