Problem: Approximate the sum of the series
by its partial sum s100 and estimate the error of this approximation.
Solution: It is easy to show using the Alternating series test that the given series converges (an almost identical example can be found in the section Convergence of general series in Theory - Testing convergence), so the question makes sense.
My calculator says that
The error of approximation is
Since the given series is alternating, the most natural way to estimate the remainder in a series is to apply the appropriate fact using bN+1.
Thus
Sometimes we also use an upper estimate by something nicer (integrable expression, geometric series) when estimating the error of approximation. Here the changing signs force us to use absolute value for such a purpose.
However, the series on the right is divergent and thus this kind of estimate does not help here.