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(a) Find the partial sum $ s_5 $ of the series $ \sum_{n=1}^{\infty} 1/n2^n. $ Use Exercise 46 to estimate the error in using $ s_5 $ as an approximation to the sum of the series.

(b) Find a value of $ n $ so that $ s_n $ is within $ 0.00005 $ of the sum. Use this value of $ n $ to approximate the sum of the series.

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a. $\approx 0.00521$b. $\approx 0.693109$ is within 0.00005 of the actual sum.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

University of Nottingham

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.

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\begin{equation}\begin…

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(a) Find the partial sum $…

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01:38

Use the result of Exercise…

Estimate the error in usin…

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01:13

Estimating errors in parti…

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Use the sum of the first 1…

they were given this infinite Siri's here and for part a Let's find as five, which is by definition is the partial, some starting at one in ending at five. So in this case, one could just go to the calculator or take some time and then just add up all these fractions and going to the calculator. I get a fraction simplifies nine sixty on the bottom. And if you want a decimal, I'll give you the exact and then we have a five for one. And then once we have the six that creepy indefinitely. So that's the first part of party here finding this five. And now we would like to use Exercise forty six to estimate the gear here. So in order to do that, let's go ahead and observe the following. So here, in our case, the common ratio are in and plus one over. And now Anne is given by this up here. So if we plug this in, here's an plus one, and then here's an so simplifying this we get this fraction here. So and now way. Want to show that this is the increasing sequence, So one way to do that is Tio. Go ahead and just define FX as such. In this case already, that should be a exploded and And all I did was I just took our in the formula and replace and with the ex. Then he refuse the question rule. You'LL get the following after you simplify and this expression is positive. So that means that f is increasing, so that means our is also increasing. And now we can go ahead and use exercise forty six. So it's for the next page. This implies the remainder when using in terms is bounded about by this expression or l is the limit of the Oran. So in our case, that will be just one half because from the previous page, that's our inn and this simplifies to one half. So here, in our case, let's go ahead and replace and with five. So we have our five and then what? E six here. So I'm getting to a six, which is to over six times to the sixth, and you could go ahead and cancel one of those two is there. And if you'd like to go to the decimal. Okay, so that resolves part, eh? now let's go on for party for party first, let's give the statement. So here we want to find n such. That sn is within this decimal here. That's four zeros after the decimal with the five of the Sun, the Infinite song. So recall from part ay that where we have the following here. So so in our case, using the formula for a N and then one minus l. That's just one half in the denominator saw all too small subside by two up top. And we want this to be less than zero point zero zero zero zero five because that's the what's being asked of part B. So here, once again, let me go on to the next page. Continue where I left off. So I'll go ahead and cancel one of those two's from the previous page, and we have this. And now it's just a matter of finding some men value that works. And in this case, after trying a few values for in, I see that if I flug in eleven, we get the following, and this is indeed less than. And so therefore with this ends is that if we want Teo approximate the sum with small here, we can go ahead and just take and equals eleven and going to the calculator once again. So that's the value then. And this would be the corresponding some, and that resolves Harvey.

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