Wednesday, October 15, 2008

Series Part 3

As promised, in this post I will shortly discuss infinite series. I don't want to say a lot about this because this would require developing precise definition, so I will just describe the main ideas and give some classical examples.

The main question we can ask about an infinite series is if it has a limit, or in a case of an infinite sum, the result of the summation.
Firstly, lets define what a limit of a series is. Simply put, a limit of a series is a single number to which the series is close from some point. For example, the limit of the series:


Is 1 (we assume that the series continues to infinity following the same pattern). The same is true is we change the first number - the limit will remain the same. In a counter example, the series: 0,1,0,1,01.... Doesn't have a limit at all, because it is wrong to say that there is single number to which the series is getting close to.
It is important to notice that the definition I gave is not exact. Moreover, if you will write such a definition on a exam you will get null point for it. It doesn't mean that it is wrong - but it is not useful. It is simple to fix, all that is needed is to define what does it mean "close to" and "from some point". Unfortuantely, the script I used to write latex in posts no longer works, so I cannot write a definition (it requires writing symbols). Therefore completing the definition is left as an exercise for the reader :).

Since we have the "definition" lets try to use it. Lets look on the series:


It is very easy to see to what number this series is getting close to. All you need to do is to imagine that n is very big (or use a calculator). Either way you will see immediately that the result is very close to zero. However, according to the definition we need to show that there is only one number that the series is getting close to. To do this lets suppose that r is another number that the series is close to. Since the series is positive, if r is negative than the series is always closer to zero than to r, so r is not the limit. Otherwise, r is positive. So lets find n such that 2^(-n)<0.25r. Because the series is monotonically decreasing, we get again that from some point the series is closer to zero than to r.

Lets look on the series: a(n)=4+n. According to our definition it doesn't have a limit because the limit must be a number and it is clear that this series grows "to infinity". (It is of course possible to define the limit differently). From this two examples we get a simple result - all arithmetic series don't have a limit and all the geometric series for which |q|<1 have a limit. Also, for all such geometric series the limit is non other that zero, because we get: a(n)=aq^n. If q is less that one, this number approaches zero.

Lets now look on infinite sums. We will say that a sum converges if the sum is a real number. (Again, the definition is not precise.) This brings a question - how is it possible at all? After all it doesn't matter how small the numbers we sum, if we sum an infinite number of them the result "should" also be infinite. At least that is what our logic tries to say. But, this is wrong. Lets see an example:
1, 05, 0.25, ....

This is a simple geometric series, in which q=0.5 and a=1. From the previous post the sum is:


We can think about this as a new series - a series of partial sums of the original infinite series. Does it has a limit? Of course - it is obvious that the limit is simply 1/(1-q). Now, if we will define the infinite sum as the limit of the series of the partial sums, we will get the immediate result - the sum of the infinite series. In the case of p=0.5, the sum is 2.

In the next (and final) post I want to discuss an example of a series that is not an arithmetical or a geometrical progression.

No comments: