Sequences and series

Consider the following sequence or list of numbers:

This is the sequence whose n-th term is 1/n. Now we say that the sequence has the limit 0 as n goes to infinity,

.

What we mean by this is that if we go out far enough in the sequence, all the terms will be as close as we please to 0. For instance, if we want to be sure that 1/n < 1/10000, all we have to do is go out beyond the 10000-th term in the sequence.

From this we get the notion of an infinite series. Consider the partial sums
1
1 1/2 = 1 + 1/2
1 3/4 = 1 + 1/2 + 1/4
1 7/8 = 1 + 1/2 + 1/4 + 1/8
...
1 2047/2048 = 1 + 1/2 + 1/4 + ... + 1/2048
and so on. Now these sums are headed somewhere, namely, toward 2. If you add on enough terms, eventually you will get as close as you like to 2. So we say that the sum of all the terms is 2, that is
.

We say that an infinite series with a sum converges. Some series don't converge, like 1 + 1/2 + 1/3 + 1/4 + ..., the harmonic series.. To see that the partial sums go off to infinity, note that we have 1 and 1/2 and then
1/3 + 1/4 > 1/4 + 1/4 = 1/2,
1/5 + 1/6 + 1/7 + 1/8 > 1/8 + 1/8 + 1/8 + 1/8 = 1/2,
the next 8 terms, 1/9 + 1/10 + ... + 1/16 > 8 * 1/16 = 1/2,
and so on. So our sum grows larger than any multiple of 1/2. We say that the harmonic series diverges.

The subject of sequences and series is full of interesting things. Here are just two:

and Wallis's infinite product for pi:

Demo of Wallis's product (requires Java support)