Note: If you can't find a suitable explanation for the whole alternating series deal, just tell me and I'll go through it. I'm just being incredibly lazy right now.
I think you're taking the wrong approach. (nevermind about this)
Let's take the negative out of r and introduce it seperately into the formula (I also changed a=1):
This is clearly an alternating series, and it satifies the Alternating Series Test for convergence, and therefore, we can utilize the Alternating Series Estimation Theorem. (Google these things if you aren't familiar with them).
...but now that I think about it, the series' values are fairly "strong" for a while, so you have to go through quite a lot of them.
Anyway, I went through them, and by the time you hit (I don't know how to make the approximate sign in LaTeX...and here I'm talking only about b-sub-n part..where n=14[again, don't know the LaTeX for it]) you're at a point less than . So we have to go up to the point n = 13 as at this point, b-sub-n equals ...which still modifies the 4th decimal place, but that's okay because the next value in the series does not.
But I don't think this approach is best....so nevermind...if you're curious how I found it out, just look up the theorems (if you didn't know them beforehand)...and I don't understand your approach at all...so I automatically think it's vastly more elegant than mine.