To determine if the series diverges or converges we need to look at its sequence of partial sums. We'll start by calculating a few partial sums so we can see what they're doing: Let's figure out the pattern so we can find the general term *S*_{n} of the sequence of partial sums. The *n*th partial sum is given by To determine if the sequence *S*_{1}, *S*_{2}, *S*_{3}, ...
converges, we need to know what happens to *S*_{n} as *n* approaches ∞. In symbols, does exist, and if so, what is it? Since we have a formula for *S*_{n} we can answer these questions. We know As *n* approaches infinity the term '-1' in the numerator becomes so small as to be irrelevant, so we're basically looking at the fraction This means Since the sequence of partial sums converges, the original series converges. Since the sequence of partial sums converges to 1, we say the sum of the series is 1: Zeno's Paradox is resolved. Although there are infinitely many fractional distances between us and the brownie, all those little distances add up to 1. The math agrees with what we already know: we can indeed get to the brownie. Hopefully, it's an infinite brownie so we never have to cross this room again. |