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Determine the number of terms required to approximate the sum of the series with an error of less than .001

Sum ((-1)^(n+1))/(n^3) from n=1 to infinity

2. Homework Equations

3. The Attempt at a Solution

I guess this is what you do:

1/(n+1)^3 < 1/1000

and solving you get n+1 > 10 so the answer is 10 terms

But that doesn't quite make sense to me, and I'm not sure why.

Alternating series remainder theorem:

|S-Sn| =|Rn|< or = to an+1

Could someone please explain this to me?