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$$\sum_1^{\infty} (-1)^{n+3}\frac{n}{n^2+8}$$

Here is my question: Using the alternating series error estimation theorem, what is the smallest number of terms needed to estimate the entire sum with an error of magnitude less than $10^{-6}$?

So from what I know about the alternating series error estimation theorem, it's when you have to list out the terms and adding up a certain number of terms makes the error of the estimation the next term that was left out of the group of terms.

So I wrote out the first few terms, is that how you approach this problem? Am I supposed to find the term with the 0.000001 place?

Here are the first few terms:

0.11 - 0.166 + 0.176 - 0.16 + 0.151

So far I do not see any term to that place. Is there some other way to do this problem?

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    $\begingroup$ I suppose that you did not finish your last sentence. $\endgroup$ – Claude Leibovici Apr 18 '15 at 8:36
  • $\begingroup$ oh wait let me fix that thanks $\endgroup$ – Elsa Apr 18 '15 at 8:37
  • $\begingroup$ So your problem is that none of the five first terms is less than 10^{-6}? Sorry but (1) does this really come as a surprise? and (2) sure you have no idea how to overcome this obstacle? $\endgroup$ – Did Apr 18 '15 at 8:44
  • $\begingroup$ @Did well um ya I just started learning about this theorem, so (1)ya it comes as a surprise and (2)my only idea is to keep listing out terms. Is that how it goes? $\endgroup$ – Elsa Apr 18 '15 at 8:46
  • $\begingroup$ Imagine that the first suitable term is at n = 10^6... Still think this is how it goes? And if you were asked for some rest less than 10^{-12}... $\endgroup$ – Did Apr 18 '15 at 8:47
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You could use the Calabrese criteria (see the references) to approximate the sum of alternating series:

References:

http://ecademy.agnesscott.edu/~lriddle/apcalculus/approxSeries.pdf

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