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JH
Numerade Educator

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Problem 33 Medium Difficulty

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

Answer

0.00000004

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Video Transcript

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.