Finally, Meaning...and Food Examples

Determine if the series

from Zeno's Paradox diverges or converges. If it converges, what is the sum of the series?

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 nth partial sum is given by

To determine if the sequence

S₁, S₂, S₃, ...

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.

Determine if the series

converges or diverges. If it converges, find its sum.

To answer this question, look at the first few partial sums:

S₁ = 1 = 1

S₂ = 1 + 2 = 3

S₃ = 1 + 2 + 3 = 6

To get from one partial sum S_n to the next, we have to step up more than 1. This means S₂ must be at least 2, S₃ must be at least 3, and so on.

Then

must be greater than 1, 2, 3,... and every other natural number n. This is impossible. In other words, the limit doesn't exist because the values S_n zoom off to infinity. Since the limit of the partial sums doesn't exist, the original series diverges.