You can estimate by computing (in general ). Your estimate is going to be a little bit low since you only computed the first 10 terms of the series. (I think that's what this is supposed to be) gives you an upper bound on how far off is from the true value of the series. If you want to be within , choose such that .
The last part says that is guaranteed to be smaller than the true value of the series, while is guaranteed to be larger than the true value of the series. So you have two estimates, one which is too small and one which is too big. If you split the difference and average the two estimates, you should get something closer than either estimate. Specifically, the error will be at most half the difference between the two estimates.