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Estimate the magnitude of the error involved in using the sum of the first four terms t0 approximate the sum of the enti (-1)n+t 2 -1 <510.20 o. /5|Click t0 sele...

Question

Estimate the magnitude of the error involved in using the sum of the first four terms t0 approximate the sum of the enti (-1)n+t 2 -1 <510.20 o. /5|Click t0 select your answer

Estimate the magnitude of the error involved in using the sum of the first four terms t0 approximate the sum of the enti (-1)n+t 2 -1 <51 0.20 o. /5| Click t0 select your answer



Answers

Estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{1}{10^{n}}$$

So for us to determine the error of an alternating Siri's, um they tell us in the book that whatever term we want to add up to this is going to be less than or equal to, um the plus one term. So over here, let's just first figure out what is a n and all that. So this negative one to the end matches up here. This A M matches up with tea. And so first we know that a N is equal to t to the end. And so what we want is first for terms. But since we're starting, our in text from in is equal to zero. That would be like 0123 So for us in is three, which then implies over here in plus one is going to be four. So we're gonna have a four is equal to t to the fourth, and then if we take the absolute value of this, this is going to be less than or equal to our error. It's absolute value on Technically, I guess we don't need the opposite values on the left side, since T to the fourth is always going to be positive. Um, but I just kind of believe it here for right now. But now we need to figure out Well, what is this? Because I really don't care so much about just saying it's t to the fourth. Well, if we want to get this absolute found, we already know that T is going to be tracked between zero and one. So actually, let me do this right above here. So if we were to just raise all sides here to the fourth power, this implies that zero is less than strictly less than, um t to the fourth, strictly less than one. So we already know that the error here has to be, um, greater than or equal to zero just because absolute value. But this now tells us that teach the fourth has to be less than one. So we know that that's less than what. And so what this implies is that our error absolute value is always going to be strictly less than one. So this gives us are bound

Let me noticing as the Kastner and we want you estimated a revision of us for autumn. She means Bender. And when us Stan the manager him a video. And this is really good. And by the serum will see that this terrible this morning Nico Juna many toe under Next term there is only with the asked for you. So to me Uh, five. Now if I were you go, Joe one off, five. And as we read the, uh I'm losing the first far terms job. Chris Matus Some mission here.

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All right. When we're estimating the magnitude of the error involved. When estimating the sum of an alternating series with partial sums, we need to remember the alternating series estimation theory. The steer um tells us that if we estimate the some of the alternating series with non alternating part, use a bent. If we estimate this with the partial sum S Sir Ben then the magnitude of the error. In other words, the absolute value of the error that is less than the numerical value of the first unused turn. If the partial some were using is S. A. Ben. That means we're using all the terms up to negative one to the end times use a bin, which means that the first unused term is negative one to the N plus one times U. Sub one plus one. So the numerical value of that first unused term is you sub and plus one. That is the estimation for the magnitude of the error. So in this problem if we are using the first four terms Then because the sum because the some of the series starts counting at an equal zero Rather than n equals one. Because of that, the partial sum we actually want is S. Three not as four because this the series some starts at an equal zero. So yeah wow. Excuse me. If S three is the partial somewhere using then the first unused term is you sub for. So the magnitude of the error, the absolute value of the error is less than you sub for. In this problem you sub for equals T. to the 4th. So the magnitude of the error is less than T. To the fourth. And in fact because T equal, I mean T is less than one. We can also say that this error ah Which is less than T to the 4th Is also less than one. So this is an estimation for the magnitude of the error involved.


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