(a) Use the sum of the first 10 terms to estimate the sum of the series ∑n = 1^∞ 1/n^2. How good

(a) Use the sum of the first 10 terms to estimate the sum of...

Key Concepts

Determining Number of Terms for Desired Accuracy

This concept involves finding a value of n such that the error between the partial sum and the actual series sum is within a specified tolerance. The process usually leverages error bounds or tail estimates to ensure that the approximation meets the given accuracy requirements.

Estimation Improvement Techniques

Beyond taking a simple partial sum, various techniques can be applied to improve the estimate of an infinite series. These techniques may involve corrections based on the behavior of the tail or using formulas that provide an enhanced remainder approximation, thereby refining the estimate of the infinite sum relative to the partial sum alone.

Error Estimation and Remainder Approximation

When approximating an infinite series with a partial sum, it is important to understand the error, which is the difference between the actual sum and the computed partial sum. Error estimation techniques, often through remainder term analysis or integral approximations, allow one to bound this error and ensure that the approximation meets a required precision.

Partial Sum Approximation

A partial sum is the sum of the first n terms of an infinite series, and it serves as an approximation of the series’ total sum. Understanding partial sums is crucial for estimating the sum of an infinite series by calculating a finite number of its terms, and analyzing how closely this finite sum approximates the actual value of the series.

Infinite Series Convergence

This concept deals with whether an infinite series approaches a finite limit and explains the conditions under which a series converges. In general, techniques and tests (like the p-test, comparison test, integral test, etc.) are used to determine the convergence of series, which is fundamental when approximating the sum of an infinite series using a finite number of terms.

p-Series

A p-series is an infinite series of the form ?1/n^p, and its convergence properties are well studied. For values of p greater than 1, the series converges to a finite value, which makes p-series an important example in analysis and helps in setting benchmarks for estimation and error analysis in series approximations.

Transcript

00:01 Okay, so for part a here, let's go ahead and use s10 to approximate or estimate the infinite sum s.

00:13 And also, we'll also ask, how good is the estimate? so here, by s10, of course, we mean the tenth partial sum.

00:34 So the solution for part a, so first we'll write out s10 by definition.

00:41 This is the sum of s just for the sum from one to 10 so in that case just go ahead and write out the few terms here and add up a bunch of fractions this would take quite a while here so one could go to the calculator just round off here let's at 1 .549768 and so that's an approximation to now to answer the second question, how good is the approximation? well, if you look at exercise 34, you'll see that they give the value for s, for the sum here, the infinite sum, given by oiler.

01:40 So in our case, the error is the difference between the exact value and the approximation.

01:51 So since we now have the exact value, we'll plug that in, and we'll also plug it our approximation, as 10.

02:05 And in this case, we're getting about 0 .0952, which is less than 1 over 10.

02:17 So it's not a bad approximation because it's less than 1 over 10, but we can do better by taking larger n.

02:29 So it's not a bad estimation considering we only use 10 terms, but 1 over 10 just might not be accurate enough depending on one of its purposes.

02:39 So let's go on to part 3.

02:40 B now for part b this is where we'll actually go ahead and improve the estimate from part a using inequalities 3 and plugging in and equals 10 into 3 so let's go ahead and solve this so solution so first let's recall from part a what we found we found that s 10 was about okay so now if you look at inequalities 3 it gives upper and lower bounds for the exact value in terms of s n and these integrals here.

03:47 So notice the difference in the lower bounds of the integrals, one's at n plus one, the other one's at n.

03:53 So here, because we're to use n equals 10, so that determines that.

03:59 And if you recall from our series, a n is 1 over n squared.

04:04 And so that means f of x should just be 1 over x squared.

04:07 So let's go ahead and plug all this in here.

04:28 Okay, so then now, we could actually just go ahead and integrate this.

04:33 We do have improper integrals here, but these ones are not too bad, because when you plug in infinity, the expression becomes zero.

04:52 And so that would become s10 plus 1 over 11, less than or equal to s, less than or equal to s10 plus 1 over 10.

05:03 So then now one would use the information from part a and plug that in for the s10 tier so plug those in for s10 and then add the fraction.

05:18 And so going on to the next page, i'll write this out.

05:25 So there's s10 plus 1 over 11 and then s10 plus 1 over 10.

05:41 And then just add those together and there we go.

05:59 Now to find a value for s here, so think of it this way.

06:03 We have an interval here, a lower and upper bound.

06:11 And all we know is that s falls somewhere in between here.

06:18 So instead of choosing either of the endpoints, another approximation would just take the midpoint of these two, the average.

06:30 And so here we'll take s to just be the average here.

06:35 So there's an approximation to s.

06:52 And this becomes also we'd like to know the error involved here.

07:04 And so there's a few ways to do this.

07:07 So this is our approximation for s.

07:11 And so since it's in the middle, the error is just the half the length of the interval...

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