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Consider the seriesof the series by using the sum of the first 10 terms: Estimate the error involved a) Approximate the sum in this approximation required t0 ensur...

Question

Consider the seriesof the series by using the sum of the first 10 terms: Estimate the error involved a) Approximate the sum in this approximation required t0 ensure that the sum is accurate to within 0.0005? b) How many ters are

Consider the series of the series by using the sum of the first 10 terms: Estimate the error involved a) Approximate the sum in this approximation required t0 ensure that the sum is accurate to within 0.0005? b) How many ters are



Answers

Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
$ \displaystyle \sum_{n = 1}^{\infty} 5^{-n} \cos^2 n $

To use the some of the first ten terms. That's us ten Eagles. The sum from one to ten. Now this sum has been evaluated already by a computer. So if you'd like you to pause the screen and write down that large fraction, or there's your decibel about point two. So now we see the Siri's right here. This conversions by the comparison test sense one over and plus hoops into the fifth plus five is less than or equal to one over and fifth. And we know this comm urges because P equals five, which is larger than one. So we're using the pee test. So here, let's go on to the next page. So the ear is less than her equals who so we are using ten terms. So we'LL have our ten, and this is less than or equal. So this is the remainder. After using ten terms, so has explained on page seven thirty. This is less than or equal to teeth him where this he is the remainder. After using the integral test and from the integral test, we know that the upper bound for but he is of this farm we got from ten are our value. Under the tea is ten to infinity and then one over excellent Fifth the ex. So this is coming from the upper bound that we used in the comparison, the one over into the fifth. So now let's evaluate that That's negative. One over four, next to the Force ten to Infinity, and we could go ahead and simplify this. So that's a zero decimal and then followed by seven more zeros after the decimal number four. So that's approximation of the ear of the ear, which is on the left side is no more than this number over here, which you can also write is four times ten to the minus eight, and that's your final answer.

Let's use the some of the first ten terms to approximate the sum of the Siri's. So on the left over here, this is the sum of the Siri's, and that's approximately equal to us. Ten as ten. We know that's just the sum after using ten terms. So let's go to my next cabin, Wolfram Alpha, where, if you can see, I approximated the S ten the some of the first ten terms, so you could pause the screen. Write down a few decibels here for this approximation. So now that's are approximate. And now we want to estimate the ear as a result of using ten terms. So they're here is that most are ten. That here, this is the remainder. After using ten terms, that's a notation in the book. Now, before we go on in this direction, let's define affects to be one over X. Excellent fourth. So notice that if it's positive here, we're looking at X bigger than or equal to one because of the starting point is one and equals one. So if this positives on his ex is bigger than your equal to one, we could see that F is continuous because the numerator and denominator Robles continuous, and then we can see that f is decreasing. So just show this part. We need to see that the derivative is negative. So here is the question rule. Then swear that denominator. So the denominator is always positive, but we can see that the numerator is always negative. We can pull out the negativity and then we're less with X squared. Plus for X Cube. That's Herman. The parentheses is positive that the nominators positive. However, this negative is always a negative number, So this thing is less than zero. That means efforts also decreasing. So here we would to show convergence for the original Siri's would use the integral test. And so we're using the upper bound for the ear that's given by the inner will test. So here the upper bound would be the integral from ten to Infinity E one over X over X for the fourth and going head and evaluating this in a computer. This is approximately point zero zero zero three five nine and then three, six four so of repeated that point zero zero zero three five nine three six four. So the ear is no larger than this decimal over here. So that's our estimate in our final answer

In this question importante so as then equals some off n equals one to 10 one over and power three, which is approximately equals 1.975 So integration from 11 toe infinity one over x or three d X equals clam it When P approaches Infinity integration from 11 Toby Export Negative three d X equals negative. Explore negative to over to with our boundaries 11 and P if we substitute by these limits, we will get 1/2 42 and the integration from 10 to infinity one over exports three. The X equals fame as before limit wimpy approaches Infinity integral 10 to be explore Negative three The X equals negative export negative to over to and our limits 10 be which is equal one over 200. So in this case, we can say that 1975 1.975 plus one over to force to is listen, S and s is this and 1.9 75 plus one over 200 so we can simplified as s is less than 1.2253 and greater than 1.20166 now importante every threat as equals, some off an equals one toe infinity one over and pour three, which is approximately equal. 1.2 166 plus 1.20 253 over to which is equal 1.202095 So we can say that error in this case is listen or equals. 1.20 to 53 minus 1.2016 six over two equals 4.435

This question here that the nurses has asked a question. Asked Joe. A person main, the magnitude of the original postpartum immense, then will be asked, minus asked. Far in Kochi are. Yeah. And so with this morning I caught you in the middle of the next time will be five. And if I've got you won over 10 about a five and this will be the answer.


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