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The meaning of the decimal representation of a number $ 0.d_1d_2d_3. . . .$ (where the digit $ d_i $ is one of the numbers 0, 1, 2, . . . , 9) is that $ 0.d_1d_2d_3d_4 . . . = \frac {d_1}{10} + \frac {d_2}{10^2} + \frac {d_3}{10^3} + \frac {d_4}{10^4} + . . . $Show that this series always converges.

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$\sum_{n=1}^{\infty} \frac{d_{n}}{10^{n}}$ will always converge by the Comparison Test.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Missouri State University

Campbell University

Harvey Mudd College

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.

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The meaning of the decimal…

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Decimal numbers Any real n…

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Consider the series $\sum_…

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Determine whether the geom…

Let's show that this some on the right hand side always converges. So here the d ies these were the digits, and they're all numbers between zero and I. So this means that the one over tens d to over ten square and so on if I just keep adding and going in this direction forever, that this is less than or equal to if I just go ahead and replace all of the DEA's with nine. So this is just by using this fact up here for D one, then we did it for D two and so on, and we keep doing it for all these. Now we see that the Siri's on the right hand side, this's geometric, and we see that there are what are we multiplying by each time? Just won over ten. So this will converge and we even know what the sum is. You take the first term of the series, and then he just divide by one minus R. So in this case, the first term is nine over ten, and then our was won over ten, so one minus one over ten. So we have nine over ten over nine over ten, and that equals one. So on the other hand, we know that D I over ten for any number. I is always bigger than or equal to zero since d eyes bigger than or equal to zero. This the reason I'm pointing that out is because if we want to use the comparison test, we need to make sure that we're on ly dealing with positive terms. It's part of the hypothesis for the hero. So we just shown that our Siri's the one on the left hand side, which I'll Circle and Blue. We wanted to know whether this converged. We bounded in above by a larger Siri's that was Geum. Measure it that converges toe one. So we know that our Siri's also converges So d one over ten D to over ten square and so on disc convergence by the comparison test. And that's our final answer

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