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JH
Numerade Educator

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Problem 41 Medium Difficulty

Show that if we want to approximate the sum of the series $ \sum_{n = 1}^{\infty} n^{-1.001} $ so that the error is less than 5 in the ninth decimal place, then we need to add more than $ 10^{11,301} $ terms!

Answer

$1.07 \times 10^{11,301}$

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Video Transcript

this problem will show that if we want to find a partial some to approximate this infinite sum here such that the ear is less than this decimal here that will be eight zeros and then a five in the ninth spot. Then we'LL show that we'LL need to take and larger than this quantity over here. So this is our am So that means we should find off of X by just replacing and with X And then here X is bigger than or equal to one. And the reason for pointing that out is if you use in terms then from the section we know that the air is bounded above and we want thiss which is equal to one. You can go ahead and replace F with the formula up here and we would like this here to be less than the given quantity. So instead of writing this each time, let me just go ahead and denote this by some water, eh? And now this will go ahead and saw friend. So that means that we should just go ahead and use the power rule here and integrate this and that inequality is equivalent to this inequality here and we could just keep rewriting this and then this is equivalent to. So now, finally, to solve this for end will raise both side to the one thousand power that'LL cancel out this here and that'LL leave us with n larger than one thousand over, eh? All to the one thousand color and going to the calculator we get and we can see here that will need more than this many terms because this larger this numbered out here is larger and that completes the proof.