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 value of $ c $ if$ \displaystyle \sum_{n = 2}^{\infty} (1 + c)^{-n} = 2 $

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

Campbell 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.

06:15

Find the value of $$c$ if …

00:55

Find the value of $c$ such…

01:52

Find all values of $c$ for…

03:21

Find the value of c such t…

04:05

Find the value of c if…

01:40

Evaluate $\sum_{n=1}^{\inf…

05:38

Let's find the value of C that makes the Siri's equal to so this Siri's here three. Right? This because it's actually geometric weaken. Right, This is one over one plus C to the end cheer measure, and then our equals one over one plus C. And we should eventually make sure that the absolute value of our which is absolute value one over one plus e. But this is less than one. So we have one over one plus c, an absolute value less than one. That means one plus C in the absolute value is bigger than one. And this tells us that we want one placide data that one or one plus see less than minus one. So we either want seed to be bigger than zero. We're over here, see less than negative, too. So we might get more than one answer. But it has to satisfy one of these, too. So let's keep that in mind. See, Has to be bigger than zero. And the reason we want to keep this in mind is in case we have more than one solution, or if even if we have one solution, if it doesn't satisfy one of these, too. For example, C equals negative. One would not be a good choice. It doesn't satisfy one of these, too. So now, assuming C is bigger than zero or seamless the negative two. So this is just to ensure that that flew value bars less than one. We know that the geometric series we have a formula for this. It's the first term of the series which corresponds to any equals two and then we divide by one minus. R. Now let's go to the next page. So the first term when you plug in and equals two and then one minus R, it's got its simplify this what? And then this becomes one over C C plus one and recall that this entire sum was equal to two from the previous page so we can rewrite this equation and then solve this for sea. So we have C equals negative one plus or minus. So this is the quadratic formula. Unless you factor that somehow. So here, B squared, minus four a c. So one, then being plus four times that's going to be a half there and then over to a so it's giving us negative one plus or minus room three over, too. So recall that the radical three is about one point seven. So we're getting approximately negative one plus or minus one point seven over, too. So that's about two point seven over, too, or oops, sorry and see the negative one there. Point seven over, too. Or if we have the minus sign and that's about it. Earth equal to so here recalled. The value of our C had to be either bigger than zero or C was less the negative, too. This value does not work, so we just want to choose the one that we got from the plus side. So here we should on Lee take oh C two b negative one plus the square root of three over, too. Because that one does work, it's point three five, and that one does satisfy one of these, too. So negative one plus room three. All over, too. That's our values for C. That's the on ly one, and that's the final answer

View More Answers From This Book

Find Another Textbook

00:01

Researchers collected information on the body parts of a new species of frog…

03:07

For the given data given in stem-and- leaf display form, find the inter-quar…

03:00

QUESTION 23Urn A has 5 white balls and 7 black balls. Urn B has 3 white …

06:18

QUESTION 3Urn A has 5 white balls and black balls; Urn B has 3 white bal…

01:58

Q. No. 2: Suppose we want to take a sample of size 3 from a population that …

02:37

point) Consider the recursionFind all equilibria, then characterize the …

04:22

Suppose that electrical components have lifetimes that are independent and c…

02:50

(j) A process sampled 20 times with a sampk of size eight resulted int…