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JH
Numerade Educator

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Problem 35 Medium Difficulty

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 $

Answer

error $\approx 0.00000006$

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Video Transcript

let's use the some of the first ten terms to approximate the Siri's. So, in other words, this entire series is approximately as ten, and that's equal to the sum from one to ten five, minus in and then co sign swearing in. Now I've approximated this in the calculator already and Wolfram, so you could pause the video, go ahead and write on a few of those decimals. So that's our approximation to the infinite sum. Now this is where we'LL go ahead and actually see what the errors. So first, the best way to go about the year is the first. Think about how you would show this thing convergence. So looking at this is less than or equal to fight to the minus end, says co sign, squared, and is less than or equal to one. And for this series, this converges. You could use the fact that it's geometric. So, for example, for the here. So this is the noted by our ten, and this's bounded above by the air that you would get from using the integral test. So this is explained in more detail on page seven thirty, and then here. If we were to use the integral test. So here this is the integral from ten to infinity, one over five decks. So here you have to show, of course, that the function one over five X So we have to show that it's positive, continuous. These air both clear, and the last one's also clear that is decreasing. It's decreasing because every time X gets bigger, the denominator gets bigger, so the fraction it smaller go ahead and integrate that that's one over five to the ex over Alan off one over five tend to infinity. And so this is approximately point zero zero zero zero three more zeros and then a six. So let's a decimal. And then after the decimal, we have seven zeros, followed by six. So this is an upper bound for an ear, so that gives us an estimate for the year. And that's our final answer.