Infinite Series

An infinite series is an expression like this

S = 1 + 1/2 + 1/4 + 1/8 + …

The dots mean that infinitely many terms follow. We obviously can’t add up an infinite number of terms, but we can add up the first n terms, like this:

S1 = 1
S2 = 1 + 1/2 = 3/2
S3 = 1 + 1/2 + 1/4 = 7/4
S4 = 1 + 1/2 + 1/4 + 1/8 = 15/8

It is clear what the pattern is: the n-th partial sum is

Sn = 2 – 1/2n

When n gets larger and larger, Sn gets closer and closer to the number 2. When a sequence Sn gets closer and closer and closer to a given number S, we say that S is the limit of the Sn‘s and we write

lim( Sn ) = S

To take a physical analogy, consider a student who is one yard from the wall of the classroom. He takes a large step to cut the distance to the wall in half. Then he takes another step to cut the distance in half again. He repeates this again and again, getting closer to the wall each time. He never reaches the wall, yet that is his limit postion. We could write

lim( Positionn ) = Wall

In our case lim( Sn ) = 2. Since this limit exists, we say that the sum of the series is 2, even though we can’t really “do the sum.”

Another example

Our first example was easy to understand because there is a simple formula for the partial sums. Now let’s look at a more difficult example.

S = 1 + 1/4 + 1/9 + 1/16 + … + 1/n2 + ….

We can compute some partial sums in an effort to see what the limit might be:

S0 = 1
S1 = 1 + 1/4 = 1.25
S2 = 1 + 1/4 + 1/9 = 1.36111…
S3 = 1 + 1/4 + 1/9 + 1/16 = 1.4236111….

This time it is not clear what is happening. The partial sums are increasing, since we get one from another by adding a positive number. But do they approach a limit? Is there a number to which they get closer and closer as we add more terms? If there is a limit what is it? Can we compute it to some modest accuracy, say one or two decimal places?