Series - Part 1

In math a series is a function from the natural number to some set B. The set B can be the real numbers but it can also be a vector space or some other set.
The most basic question we can ask about a series is finding the generating formula. Such formula can be extremely difficult to find, and since a series of random number is by all means a series such a formula may not even exist. Lets start with a few simple series:

Arithmetic progression:
This is probably the simplest example. An arithmetic progression is obtained by adding a fixed number to the previous number in the progression. For example, if we choose to start with the number 5 and to add seven, we will get:

5, 13, 20, 27, 34.....

In the general case we get:

a1=a, a2=a+d, a3=a+2d, a4=a+3d,...

It is very simple to find the generating formula for this progression. To do this we need to look on the difference:

a2-a1=d
a3-a2=d
.
.
a(n)-a(n-1)=d

If we add all of the equations we will get the generating formula:

a(n)-a1=(n-1)d

This method of looking on the difference can be used for many series. For it to work, we need to know how to sum the right side of the equation. In the example it was very simple to do, but usually it is much more difficult. Also, since the differences are in fact a series of its own, we can try to find the formula for the difference, and from it get the formula to the original series.

Geometric progression:
A geometric progression is a progression in which the ratio of the successive terms is fixed, that is a(n+1)/a(n)=const for all n. For example:

1,2,4,8,16....

And the general progression:

a,aq,aq^2,aq^3,aq^4...

In this case, finding the generating formula is trivial. Just from looking on the progression we get that:

a(n)=aq^(n-1)

However, if we would try to find this formula by looking on the difference we would just get the original progression back. Therefore to get the formula this way we would need to know the sum of the geometrical progression.
In the next post I will develop the formulas for the sum of both arithmetical and geometrical progressions.