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JH
Numerade Educator

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Problem 28 Easy Difficulty

Approximate the sum of the series correct to four decimal places.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {( - 1 )^{n+1}}{n^6} $

Answer

0.9856

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Video Transcript

approximate this sum up to four decimal places. Now the first thing is, I see that the Siri's is alternating and therefore I should identify the beyond the positive part of the term in this case, just one over into the sixth since since alternating I know by the theorem in section eleven point five. This alternating Siri's estimation there, um that they ear is bounded above when the heroine, using and in terms, is bounded above by being plus one. So this means we want the smallest end. Such that bien is less than five times will hear will do ten to the minus five. So this is the last number that I just pulled out here. This is just to ensure that were correct. That's a four decimal places. Two. So here, let's go in and solve that for end. So we know we have one over into the six less than five times ten to the minus five. So here we could go to a copulate or just do trial in here. But the first end that makes this true is an equal six. So we need at least six. So this tells us that we need toe Add just five terms because again, this is coming from the difference. Here was the BM plus one and the B end. So the ear is SN, but over here we have in plus one. So the n plus one is six hoops and plus one equals six implies and equals five. So going to the next page, the entire song, oops and plus one over into the six is approximately equal to the sum when just using five does Mel's, that's s five. And going into Wolfram, we just go to the four decimal places here and we have nine, eight, five, and then round up one that will just give us six, nine, eight, five, six. So that is our answer that is correct up to four decimal places, and that's our final answer.