Understanding the Mathematics of Personal Finance

SUMMATIONS

This topic is not necessary for understanding the rest of the book. It’s an introduc­tion, however, to a powerful notation that allows us to work with numbers from arbitrarily large lists in a very concise manner.

Looking at Table 1.3 again, suppose that I want to add up the values of e from e5 to e8. That is, add up e5 + e6 + e7 + e8. Just for the record, I don’t know why I’d ever want to do this with a Life Table. There are cases, however, such as adding up the interest payments on a loan, where I often want to do this.

The shorthand notation involves the use of the uppercase Greek letter sigma:

8

I e,

i=5

How to read this: Beneath the sigma you see “i = 5.” Above the sigma you see “8.” This means that we want to add up all the terms to the right of the sigma for i = 5, 6, 7, and 8: [3]

For example, Table 10.1 has 101 rows of numbers representing some informa­tion from age = 0 to age = 100. To show that I want to add up all the e values in this table, I write

100

X e,

i=0

This can be extended to more complicated expressions, such as X(ei + 7) = (e5 + 7) +(e6 + 7) + •••

i=5

The variable і is sometimes called the “index.” Note that it can also be used directly in the expression to be evaluated:

8

X iei = 5e5 + 6e6 + 7e7 + 8e8.

i=5