Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

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 {(x - 2)^n}{3^n} $

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Missouri State University

Idaho State University

Boston College

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.

01:58

Find the values of $x$ for…

01:30

00:37

03:17

Find the values of $ x $ f…

00:59

Find all values of $x$ for…

03:34

Let's find the values of X for which the Siri's converges and for these values of X, will actually go ahead and find some as well. So let's break this into two parts. First, we just want to know the X values for which it converges. So let's call that part one and part two's over here when we find the sum. So let's just rewrite this Siri's This is actually a geometric series may be in disguise the way that a certain right now I'Ll just pull out that three and then put the end on the outside of the parentheses. So we have geometric and we see that are equals X minus two over three. And we know that geometrics on ly converge when the absolute value of our is less than one. So we need the absolute value our which, in our case, absolute value of X minus two over three. You simplify that a little bit. Oh, we need that to be less than one due to this over here. So solve this for XO. First, multiply the three over and then using the definition of the absolute value. This means the X minus two satisfies this and then add the tutu all sides of the inequality. So in this case, all three sides and we have our interval for X. So these are the X values for part one for which the series will converge. And we get this from using the fact that the series is geometric and we know geometric on ly ca merges when the absolute value ours lesson one. All right, so that's enough for part one, not for part two geometric series. We know the sum will equal the first term of the series. This is always the formula. The nice thing about this formula here, the way that his friend is it doesn't depend on what the starting point is. So here we're just going we just go to the first term by plugging in the smallest. And that we see in this case is zero. So you have X minus two to the zero power over one minus R. In this case, we know where arias that's just X minus two over three. So we have let me come down here. We have one of top and then one. Let's go ahead and get that common denominators. Three minus X minus two. But in the denominator, I have another three down there. So let me put that three on the top and then let's go ahead and simplify. We have three and then plus two, which is five and then minus X. So for the answer for part two again, we're only assuming that we're only looking at these exiles betweennegative one and five. So assuming exes in there than the value of the summation that we were supposed to evaluate is equal to three over five minus X, and that's your final answer.

View More Answers From This Book

Find Another Textbook

02:01

'The following transactions were completed by the company:The owner…

02:35

"Evaluate the following integral using integration by parts4x sinx …

04:00

'Given the following propositions, p: Itis below freezing: q: Itis snow…

02:27

'FT = "12 1412 The infield ofa baseballfieleHis -& square with…

01:09

'In which quadrilaterals are the diagonals perpendicular to each other?…

01:34

'6.3.22Question HelpThe perimeter of parallelogram ABCD is 82 c…

04:06

'Find an equation for the collection of points for which the distance t…

03:59

'golf ball manufacturer knows the cost associated with various hourly p…

03:36

'Savannah owns a food truck that sells tacos and burritos. She only has…

02:10

'As shown above, a classic deck of cards is made Up of 52 cards, 26 are…