I have the sum
$$S=\sum_{n=1}^{\infty}(-1)^n(\frac{n}{n^2+1})$$
And Im trying to find the sum $S_n$ where the error approxomating $S$ is less than $\frac{1}{1000}$.
I calculated it at found what I tought was the right answer, but it turned out to be wrong. Im not even sure if Im using the corracly formulas anymore. What the right way of dealing with sum/error problems like these?
For an alternating series , the sum will converge to a given accuracy when Abs[a(n)] < epsilon. Please, have a look at http://en.wikipedia.org/wiki/Alternating_series. So, you must find "n" such that n / (n^2+1) < 0.001. Solve for "n" such that n / (n^2+1) = 0.001, get "n" and add 1. What you also can notice is that n / (n^2 + 1) is very close to 1 / n. Then n = 1 / 0.001 = 1000 is an obvious answer.
It is an alternating series and so the error between your partial sum and the true sum cannot be more than the first term you have ommitted from your partial sum. And so set the equation from the series equal to 0.001 and solve for n. If n is "fractional", which it probably will, round up appropriately. Hereby you have determined how many terms you have to include in your partial sum Sn
For an alternating series, like this one, the error is less than the first neglected term. You can just keep adding terms until the next is less than $\frac 1{1000}$ and declare success. You may have gone farther than necessary, but that is OK.
