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Let $ x = 0.99999 . . . . $(a) Do you think that $ x < 1 $ or $ x = 1? $(b) Sum a geometric series to find the value of $ x. $(c) How many decimal representations does the number 1 have?(d) Which numbers have more than one decimal representation?

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(A). $x=0.9999 \ldots .=1$(B). $\frac{a}{1-r}=\frac{0.9}{1-0.1}=\frac{0.9}{0.9}=1$(C). The number 1 has two decimal representations, $1.00000 \ldots$ and $0.99999 \ldots .$(D). 0.5 can be written as $0.49999 \ldots$ as well as $0.50000 \ldots$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

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.

06:53

Let $x=0.99999 \ldots$…

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Let $ x = 0.99999 . . . . …

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If $|\mathrm{x}|<1$, th…

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If $|a|<1$ and $|b|<…

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a. Show that the infinite …

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The meaning of the decimal…

Suppose the sum of an infi…

00:51

let x b zero point nine basically zero, followed by infinitely many nines. Let's use this shorthand notation to represent that for part a. Do you think that X is less than one or X equals one? So I know the answer already so But let's just make a guess here, make your own guests and then record it and then we'LL come back to that guests. So here we'LL end up seeing that it is exactly one which is quite surprising to most people because it does have a zero as the on the left of the decimal. Well, we'LL see this and party. Let's go ahead and prove it using party So some the geometric series to find the value backs. So here all they're saying is, Let's first right x as a sum and so on waken right This is nine over ten nine over one hundred nine over a thousand and so on, and now it's starting to appear that we have achieved metric Siri's and we know the sum of the geometric series. The sum equals the first term over one, minus the common ratio R This is for geometric son. So the first term we see that's nine over ten, and then we divine by one minus. And what is our value of our What are we multiplying by each time it's one over them, So we have nine over ten, divided by nine over ten, and that's exactly one. So that verifies that the guests, in part it was correct. And then see how many decimal representations that is the number one have while we have one death, more representation of this form. Just write it that way, however we just saw in the party here that we can also write is if you want just one point zero with a bar. So here's these air two distinct decimal representations of the same rule number. So to that will be our answer for party. And now let's generalize this to other real numbers. Let's go to a party. Open it Party's asking, which, when do two real numbers have the same that small representation? So let's answer this. The only real numbers with two decimal representations I have, they're decibel representations, agree up to some point, then one of them continues with, So this is after they agree, up to some point and then after that, after that point, one of them looks like a and then followed by infinitely many nines. And the other one, it is the other one. So this is the part in which they agree this these blank spots right here. That's the part of which there they agree, up to some point. And then that number is the first time that they that they split up and the other one instead of a it goes into some other number be followed by infinitely many zeros. And here be is just a plus one. So, for example, where is Sohn? Example? On part. See five. Here's another one. So these numbers agree up to six. And then after that we have a year, which is seven, followed by infinitely many nines. And then here we have B, which is seven plus one and then followed by infinitely many zeros. And in part, a. We saw that one point zero equals zero point nine. So here thes numbers agree up to some point, but technically they don't have to agree at all. I mean, but it is possible is we saw in this example here underline and blue. Now agreed. It is possible for them to match up for a few decibels. Then you have a point in which this witches, But if they don't agree up to a certain point, that's also fine as well as Long is, well, here's our A. Then you have a plus one. And then the days followed by infinitely many nines, the bees followed by infirmity zeros. So this is the only time in which a number will have to representations, and that's the final answer.

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