Return
NOTES

Patterns of Infinity

[-Geometric Sequences-] [-Continued Fractions-] [-Other Infinite Sums-]
We have already looked at formulas for finding the sum of arithmetic sequences (see notes on Summing Sequences) and how to find the formulas from tables (see notes on Formulas from Patterns). Here are some more formulas which have to do with infinite (or at least large) sequences. In particular, we will look at how to sum sequences like:
Geometric Sequences
Sequences of the type 2 , 6 , 18 , 54 , ... where consecutive terms have a common ratio ( r ) are called geometric sequences. In this case r = 3. The sum of these terms is infinite. That is, as we add the next term, the sum just gets larger.
There are geometric sequences though which have |r| < 1. ( that is r is between -1 and 1 or fractional )e.g.

In this case the common ratio is 1/2 .

If we now sum this sequence as in

it is not so clear what is happening when we add the next term. Does the sum grow indefinitely or is there a limit? Because now when we add the next term we are adding a progressively smaller fraction.

One way to picture this sum is to think of a length of string 1 metre long.
  • Cut the string in half and put one of the halves stretched out on a table.
  • We have 1/2 the piece of string on the table.
  • Get the remaining half and cut it in half so that we have two quarters and put one of the quarters at the end of the (1/2) string on the table.
  • We now have 3/4 of the string on the table.
  • Now halve the remaining quarter string so that we have two eighths and place one of the eighths at the end of the string on the table.
  • We now have 7/8 of the string on the table.
  • Continue this process forever. What length of string do we eventually get on the table?
  • 1 whole piece!

We call such a sum of a geometric sequence where |r| < 1 a convergent sum, because as we add the next term the sum converges to a number.

There is a very simple and beautiful formula which can give the result of such a convergent sum (where |r| < 1 )!

Pause a while to consider how elegant this formula really is - it harnesses infinity , quantifies the unquantifyable, makes the invisible visible - beautiful.

This formula can help us to get an insight into the number 0.999999...

Continued Fractions
A continued fraction is a complex fraction which has the following structure:

That is - a number plus a fraction whose denominator is a number plus a fraction whose denominator is, in turn, a number plus a fraction, and so on and so on.
Continued fractions (like the one above) which eventually stop are called finite continued fractions. All finite continued fractions can be written as a rational number (i.e. a number of the form p/q where p and q are integers and q doesn't equal 0).
To do this you start at the bottom and work up. Writing the example above as a fraction we do the following steps:


There are other continued fractions (like the one below) which never stop. These are called infinite continued fractions.



Like in the sum of convergent geometric sequences each new addition in the infinite continued fraction is a smaller fraction, so we shouldn't be surprised to find that these are convergent also.
Some of these, where the same fractions are repeated (like the one above) are relatively easy to sum with the help of some elementary algebra.

The key is to replace the repeating infinite part with a variable, let us say - x.

Other infinite Sums
We can use the same strategy which we used to sum repeating infinite continued fractions (above) to sum other repeating infinite sums. Sums such as

BACK