Geometric Series Part II

Because timesing is a funny operation, a scaling up or down operation - repeated addition or however you like to look at it, it does strange things to series.

A common ratio of 2, like in the previous post, gives you a series which just keeps getting more and more massive, into infinity. It therefore has no sum - because it just keeps going on. You can find the sum of the first n terms of course.

But what if you had a common ratio of 1? Well then your series stays exactly the same

2,2,2,2,2,2,2,2,2,.....into infinity. It's the same sort of thing as above. Finding a complete sum of the series isnot going to be possible, because there will simply be an infinite load of 2s.

But if your common ratio is a fraction:

1/2, 1/4, 1/8, 1/16

Here the common ratio is 1/2. The terms of this sequence get smaller all the time, but will never reach 0.

When we think of the sequence properly as a series:

1/2 + 1/4 + 1/16 + 1/32 + 1/64

Then we can see that it is an infinite series, and will go on halving ad infinitum.
If we add the first few terms, we get 1/2, 3/4, 13/16, 27/32....

The sum is getting larger each time but by progressively smaller amounts.

The sum is getting closer and closer to 1, but without ever quite reaching it. If it had an infinite number of terms, then its sum would indeed reach 1. Its sum to infinity is 1.

It converges on 1, and is therefore called a convergent series.

This happens with certain types of common ratio, which we will come back to.

Geometric Series

These are a bit like arithmetic series but instead of there being a common difference between terms there is a common power-type difference.

Consider 2,4,8,16,32,64...

This series appears to double each time, which means that there is no common number which gives you the nth term when added to the n-1th term.

But the common difference is "doubling" - or rather, powers of 2.

The series goes 2¹, 2², 2³, 2⁴.....

With geometric series, therefore, we don't talk about a common difference, but a common ratio - the thing you times each term by to get the next one. So this series has common ratio 2.