Series is a powerful tool in finance, among other fields. Banks use series to calculate loans, savings and investments, as well as the values of cash flows. By understanding the basic principles of series, you will gain more insight into the inner workings of finance.
Theory
A series is a sequence of numbers where you exchange the comma with a plus or a minus. Series generally look like this:
Here, is the number of the term and is the actual number in that term.
When you calculate the sum of a very long series, it can get pretty tiresome to write out the entire series. For that reason, mathematicians have managed to find a way to do this much easier by introducing the Greek letter sigma: . You can write the sum of a series in the following way:
Theory
The sum of the first terms of a series of numbers can be written in the following way:
Here, is the sum of terms. tells you that you will begin counting from term number 1, shows you which term to stop at, and is the formula that describes term number .
Series do not have to be finite. There are also infinite series. When working with infinite series, the most frequent question is: what happens to the sum of the series? Will the terms of the series become so small that in the end, no matter how many terms you add, the sum will still become a specific number, or will the terms of the series be so large that their sum becomes infinitely large? Mathematically, these two cases are called convergence and divergence respectively. In general, you have:
Theory
Convergence:
The sum of the series tends towards a specific number when .
Divergence:
The sum of the series does not converge towards a specific number, often because it tends to when .
Example 1
You have a series where
In order to find the sum of the first ten terms, you need to use the summation symbol with and . That will give you
Example 2
You have a series where
In order to find the sum of the first five terms, you need to use the summation symbol with and . That will give you
In order to check what happens with the sum when , it can be worth taking a look at the formula for the terms. In this case .
These terms become smaller and smaller the further out in the series you get.
The fact that the terms become smaller and smaller does not necessarily mean that the sum of the series converges, but it means that there is a chance that the sum of the series converges.
You will see how to decide whether it converges or not by looking at the entry about geometric series. In this particular case, I can tell you that this series decreases fast enough for the sum of the series to converge towards a number, which happens to be exactly 1. Then you can conclude that the sum of the series converges when . Mathematically, it will look like this: